Separable Differential Equations Problems And Solutions

Click here to show or hide the solution Elementary Differential Equations. solutions not included in a general solution). The Bernoulli equation 5. A ﬁrst order diﬀerential equation is an equation of the form F(x,y,y0) = 0. Separable equations A separable equation is a rst-order di erential equation where dy dx can be written as a product of a function of x multiplied by a function of y. Thegeneral solutionof a differential equation is the set of all solutions. First-Order Equations for Which Exact Solutions Are Obtainable. In 1-8, classify the equation as separable, linear, exact, or none of these. Separable Equations. This may be due to some simplification they chose to do, but I am Separable Equations Problems? | Physics Forums. It is the same concept when solving differential equations - find general solution first, then substitute given numbers to find particular solutions. The product of - k and P is negative, so the rate will come out negative like it's supposed to. What are Separable Differential Equations? 1. Separable differential equations: This is right out of the first day of an ODE course. Determine the interval(s) (with respect to the independent variable) on which a solution to a separable differential equation is defined. Final Exam Resources: Final Review. 5 Additional Methods 1. Separable Differential Equations Introduction. , Ordinary Differential Equations, ODE, DEs, Diff-Eq, or Calculus 4). Engineering: Industry Solutions. All you need do is to integrate both sides of the equation dx f(x) = dy g(y). Answers to Linear First Order Differential Equation Problems19. You should note that singular solutions are real solutions and are just as natural as the general solution. differential in a region R of the xy-plane if it corresponds to the differential of some function f(x,y) defined on R. Plant Modeling for Control Design. Dependence of Solutions on Initial Conditions. Separable Equations. Introduction. This method is only possible if we can write the differential equation in the form. 5 10 Differential equations 101Solutions of elementary and separable from MATH 16B at University of California, Berkeley. 1 Differential Equations and Mathematical Models 1. The equation three, okay, this expression down there is a general solution to the given differential equation, separable differential equation, y prime is equal to G x times h of y, okay. These are the lecture notes for my Coursera course, Differential Equations for Engineers. Step-by-step solution:. ln y = ln x+C = ln x+ln k (ln k = C = constant) i. A separable differential equation is a common kind of differential equation that is especially straightforward to solve. What is the solution of the differential. Second, since y D mz solves the Bernoulli differential equation, we have that ady D a. is a 3rd order, non-linear equation. A Note on Non-Separable Solutions of Linear Partial Differential Equations ALOKNATH CHAKRABARTI Department of Applied Mathematics, Indian Institute of Science, Bangalore- (India) Submitted by W. In this paper, we study a new class of fractional nonlinear impulsive stochastic integro-differential equations with infinite delay in separable Hilbert spaces. It is solved by first cross multiplying and then integrating each side. So € y(0)=5 and dy dt = rate at which salt goes in − rate at which salt goes out. Differential Equation of first Order and first Degree OF A differential equation of the first order and. The different is located at the number of its independent variable. Find and classify critical points of an autonomous differential equations 2. Chapter 2 Ordinary Differential Equations To get a particular solution which describes the specified engineering model, the initial or boundary conditions for the differential equation should be set. A separable differential equation is any differential equation that we can write in the following form. So the final solution of. 1 Differential Equations and Mathematical Models 1 1. Looks like ‘homogeneous but for constant’ but is ‘almost separable’ 3. A function y=ψ(t) is a solution of the equation above if upon substitution y=ψ(t) into this equation it becomes identity. First Order Equations Separable Ry0(x) = f(x)g(y) dy is a solution of the homogeneous equation, is a solution of the homogeneous problem. Chapter 1 in Review. List of problems from Separable ordinary differential equations. Solve a separable differential equation. FIRST-ORDER DIFFERENTIAL EQUATIONS. Separable differential equations are equations that can be separated so that one variable is on one side, and the other variable is on the other side. Under reasonable conditions on φ, such an equation has a solution and the corresponding initial value problem has a unique solution. This calculus video tutorial explains how to solve first order differential equations using separation of variables. Solve separable di erential equations and initial value problems. (f) You cannot separate the variables here. The reason we care about separable differential equations is that: Separable differential equations help model many real-world contexts. Differential Equations Separable Equations Arcsine. Exact Equations and Integrating Factors. 2 Solutions to Separable Equations 31 2. Using a calculator, you will be able to solve differential equations of any complexity and types: homogeneous and non-homogeneous, linear or non-linear, first-order or second-and higher-order equations with separable and non-separable variables, etc. Find the solution of y0 +2xy= x,withy(0) = −2. We may solve this by separation of variables (moving the y terms to one side and the t terms to the other side),. at each nodal point = , =1,2,3,…, , we obtain a system of algebraic equations given by (7) which is not easy to solve analytically. Homogeneous System Solutions with Distinct and Repeated Eigenvalues. Differential Equations and Linear Superposition • Basic Idea: Provide solution in closed form • Like Integration, no general solutions in closed form •Order of equation: highest derivative in equation e. You will probably see problems very like these in your own differential equations course, the first 1 here is the classic mixing problem and there are sort of 2 forms this there is either air or water. Hence the derivatives are partial derivatives with respect to the various variables. Differential Equations as Mathematical Models. Suppose we are interested in ﬁnding a similar diﬀerential equation. In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation. Initial-Value Problems. Step–by–step solutions to separable differential equations and initial value problems. Let's start things off with a fairly simple example so we can see the process without getting lost in details of the other issues that often arise with these problems. Mixing problems are an application of separable differential equations. Separable Variables. Determine the interval(s) (with respect to the independent variable) on which a solution to a separable di erential equation is de ned. 1) dy dx = e x − y ey = ex + C y = ln (ex + C) 2) dy dx = 1 sec 2 y tan y = x + C y = tan −1 (x + C) 3) dy dx = xey −e−y = x2 2 + C 1 y = −ln (− x2 2 + C) 4) dy dx = 2x e2y e2y 2 = x2 + C 1 y = ln (2x2 + C) 2 5) dy dx = 2y − 1 ln 2y − 1 2 = x + C 1 y = Ce2x + 1 2 6) dy dx. Here we will consider a few variations on this classic. In the present section, separable differential equations and their solutions are discussed in greater detail. Differential Equations Separable Equations Arcsine. To solve such an equation, we separate the variables by moving the 's to one side and the 's to the other, then integrate both sides with respect to and solve for. Example Use the main idea in the proof of the Theorem above to ﬁnd the solution of the IVP y0(t)+ y2(t)cos(2t) = 0, y(0) = 1. 1 Differential Equations and Mathematical Models. Solutions by Substitutions. Determine the interval(s) (with respect to the independent variable) on which a solution to a separable di erential equation is de ned. The Bernoulli equation 5. How to solve exponential growth and decay word problems. Trench Andrew G. other solutions] What is the domain of a solution to an initial. Elementary Differential Equations and Boundary Value Problems 9th Edition answers to Chapter 2 - First Order Differential Equations - 2. In this paper, we study a new class of fractional nonlinear impulsive stochastic integro-differential equations with infinite delay in separable Hilbert spaces. Almost separable 2. 1 A First Look at Differential Equations. If you're seeing this message, it means we're having trouble loading external resources on our website. Differential Equations Calculators; Math Problem Solver (all calculators) Differential Equation Calculator. When reading a sentence that relates a function to one of its derivatives, it's important to extract the correct meaning to give rise to a differential equation. 50): Typical differential equation: ( ) ( ) ( ) ( ) p x u x g x dx du x + = (3. Homogeneous Equations. 3) can be rewritten as y0 = −(2y −1)2. Differential Equations Calculators; Math Problem Solver (all calculators) Differential Equation Calculator. In this example we touched two problems which are very important in the study of diﬁerential equations: the representation of the solution and uniqueness of the solution. org are unblocked. We will now look at some examples of solving separable differential equations. Solve and analyze separable differential equations, like dy/dx=x²y. Competence in solving first order differential equations employing the techniques of variables separable, homogeneous coefficient, or exact equations. Suppose we are interested in ﬁnding a similar diﬀerential equation. But let's go to what I would argue as the simplest form of differential equation to solve and that's what's called a Separable. speciﬁc kinds of ﬁrst order diﬀerential equations. Solve the Ordinary Diﬀerential Equation y′ +y2 sinx = 0. We may solve this by separation of variables (moving the y terms to one side and the t terms to the other side),. The order of a differential equation is the largest derivative involved. Advanced Math Solutions - Ordinary Differential Equations Calculator, Bernoulli ODE Last post , we learned about separable differential equations. Implicit Solutions of Separable Equations; Constant Solutions of Separable Equations; Differences Between Linear and Nonlinear Equations; A first order differential equation is separable if it can be written as \[\label{eq:2. Separable Equations The Simplest Differential Equations Separable differential equations Mixing and Dilution Models of Growth Exponential Growth and Decay The Zombie Apocalypse (Logistic Growth) Linear Equations Linear ODEs: Working an Example The Solution in General Saving for Retirement Parametrized Curves Three kinds of functions, three. 18 Problems: Heat Equation 255 5. 2 Integrals as General and Particular Solutions 10 1. and Archimedes solved complicated problems through simple pictures. This might introduce extra solutions. Click a problem to see the solution. By denoting q(y)=1 h(y), we write the equation in the form. Then we say that the variables are separable in the differential equation. Differential Equations as Mathematical Models. 6 Substitution Methods and Exact Equations. Modeling with Systems; The Geometry of Systems. 8 Dependence of Solutions on Initial Conditions 102 2. pdf for a comparison between this and other methods. This video contains plenty of examples and practice problems of finding the general solution of homogeneous differential equations including solving the initial value problem. The first type of nonlinear first order differential equations that we will look at is separable differential equations. Finally, you will learn how differential equations can be used to describe physical phenomena, and you will use your knowledge of differential equations to analyze these phenomena. 4 Separable Equations and Applications 32 1. (a) On the axes provided, sketch a slope field for the given differential equation at the eight points indicated. Separable Variables. Models of Motion. This section will primarily be focusing on first order, separable differential equations. DIFFERENTIAL EQUATIONS WITH BOUNDARY-VALUE PROBLEMS, 9E, INTERNATIONAL METRIC EDITION strikes a balance between the analytical, qualitative, and quantitative approaches to the study of Differential Equations. If G(x,y) can be factored to give G(x,y) = M(x)N(y),then the equation is called separable. Equilibrium Solutions. Could anyone help me give the analytical solution of the following equations? $$2\frac{da}{dt}-a-\frac{a^5}{16}=0$$ Stack Exchange Network Stack Exchange network consists of 175 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. , the solution is unique. Because of that they are very popular problems in all differential equations courses basically in every college in the world. I understand that it works in the sense that the solutions it finds are consistent with the differential equations, but how do we know that the solutions couldn’t be. (i) where f 1 and f 2 being functions of x and y only. 3 Exercises ¶ 1. From here, substitute in the initial values into the function and solve for. An example of a di ﬀerential equation that is not separable is dy dt = t2 +y3 because there is no way to write t2 +y3 in the form g(t)h(y). Plenty of examples are discussed and solved to illustrate the ideas. (Last Updated On: December 8, 2017) This is the Multiple Choice Questions Part 1 of the Series in Differential Equations topic in Engineering Mathematics. Exercises 7. Competence in classifying differential equations as to ordinary, partial, linear, non-linear, order and degree, and to construct differential equations under given conditions. You will also learn techniques for obtaining information about the solutions of equations which cannot be solved analytically. There are two methods that can be used to solve the solution to a separable differential equation. Solved Problems. This equation can be rearranged to. Blog Job Hunting: How to Find Your Next Step by Taking Your Search Offline. These worked examples begin with two basic separable differential equations. The calculator will find the solution of the given ODE: first-order, second-order, nth-order, separable, linear, exact, Bernoulli, homogeneous, or inhomogeneous. Ifyoursyllabus includes Chapter 10 (Linear Systems of Differential Equations), your students should have some prepa-ration inlinear algebra. Separable Differential Equation. For example, the differential equation below involves the function \(y\) and its first derivative \(\dfrac{dy}{dx}\). 2 Separable Variables. 4 Separable Equations and Applications. First-Order Diﬀerential Equations and Their Applications 1 1. 50): Typical differential equation: ( ) ( ) ( ) ( ) p x u x g x dx du x + = (3. Describe the behavior of a solution to a separable di erential equation as the independent. If we can symbolically compute these integrals, then we can solve for. The solution of these types of DE is discussed in the study guide: Linear First Order Differential Equations. 1: Eigenvalues and Eigenvectors Separable equations can be solved by coefficients in a differential equation, the. It is now time. , Seventh Edition, c 2001). 3 Models of Motion 44 2. Chapters 2, 3, 6 - First-Order Equations and Applications: Solution techniques for linear, separable and exact equations. De nition 1. separable equation as we did in Example1. A ﬁrst-order linear differential equation is one that can be put into the form where and are continuous functions on a given interval. The general solution is discussed and examples with detailed solutions are presented. At this present time, we will not be concerned as to whether these solutions are always valid for all of $\mathbb{R}$. A partial differential equation (PDE)? What is the order of a differential equation? What is a linear first order differential equation? What is a separable differential equation? How can you find the general solution to a separable differential equation? [1. Use the initial conditions to determine the value(s) of the constant(s) in the general solution. This is separable, hence we can solve for z as a function of x & constantes. The initial value problem in Example 1. First Order Differential Equations. Let y be the amount of salt in the tank after t minutes. You are going to try get all the y’s on one side of the equation and put (dy) over there and then you try to get all the x’s on the other side of the equation, put the (dx) over there. A calculator for solving differential equations. Solve a first-order linear differential equation. This note describes the following topics: First Order Ordinary Differential Equations, Applications and Examples of First Order ode's, Linear Differential Equations, Second Order Linear Equations, Applications of Second Order Differential Equations, Higher Order Linear Differential Equations, Power Series Solutions to Linear Differential Equations. 6 Substitution Methods and Exact Equations 57 CHAPTER 2 Mathematical Models and. 4 Derivatives, Integrals, and Products of Transforms several. Stiffness is an efficiency issue. Undetermined Coefficients for Matrices. is a 3rd order, non-linear equation. Separable differential equation Why is there. 2 Weak Solutions for Quasilinear Equations 5. Solving Separable Differential Equations • When solving for the general solution, have we found all solutions? • What is the domain of a particular solution? Example: dy y2 dx = By separating variables and integrating, we find the general solution is 1 y x C − = +. Elementary Differential Equations and Boundary Value Problems 9th Edition answers to Chapter 2 - First Order Differential Equations - 2. Differential Equations as Mathematical Models. Exact Equations and Integrating Factors. Could anyone help me give the analytical solution of the following equations? $$2\frac{da}{dt}-a-\frac{a^5}{16}=0$$ Stack Exchange Network Stack Exchange network consists of 175 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. There's another way to write a differential equation for the situation where P is decreasing at a rate proportional to P. Initial Value Problem An thinitial value problem (IVP) is a requirement to find a solution of n order ODE F(x, y, y′,,())∈ ⊂\ () ∈: = =. It is the same concept when solving differential equations - find general solution first, then substitute given numbers to find particular solutions. Be able to find the general and particular solutions of separable first order ODEs. A vast amount of research and huge numbers of publications have been devoted to the numerical solution of differential equations, both ordinary and Partial (PDEs) as a result of their importance in fields as diverse as physics, engineering, economics, and electronics. General Solution of a Differential Equation. You will also learn enough of the general qualitative theory of differential equations to know what kind of differential equation (ordinary or partial, linear or nonlinear, separable or nonseparable, and so on) you are looking at, so that when you are solving a problem which has arisen in physics, engineering, chemistry, or any of the other. Solve a first-order linear differential equation. Differential Equations and Linear Algebra, 1. This example problem uses the functions pdex1pde, pdex1ic, and pdex1bc. To solve such an equation, we separate the variables by moving the 's to one side and the 's to the other, then integrate both sides with respect to and solve for. A differential equation is an equation that contains both a variable and a derivative. SAMPLE APPLICATION OF DIFFERENTIAL EQUATIONS 3 Sometimes in attempting to solve a de, we might perform an irreversible step. Separable Equations. The equation so obtained is the desired differential equation. Step 2: Integrate both sides with respect to x. So by convention, the solutions of differential equations are defined on one single interval. We'll also start looking at finding the interval of validity from the solution to a differential equation. First Order Equations Separable Ry0(x) = f(x)g(y) dy is a solution of the homogeneous equation, is a solution of the homogeneous problem. For most applications, the two kinds of solutions suﬃce to determine all possible solutions. This method involves multiplying the entire equation by an integrating factor. }\) This technique allows us to solve many important differential equations that arise in the world around us. Step 4: The initial condition means when. In this section, we focus on a particular class of differential equations (called separable) and develop a method for finding algebraic formulas for. Differential Equation of first Order and first Degree OF A differential equation of the first order and. Differential Equations and Linear Algebra, 1. The simultaneous solution of these equations is a = 3 and b = 1. Differential equations are mathematically studied from several different perspectives, mostly concerned with their solutions -the set of functions that satisfy the equation. Introduction to differential equations View this lecture on YouTube A differential equation is an equation for a function containing derivatives of that function. differential equations have exactly one solution. Lectures on Differential Equations. This equation can be rearranged to. Coupled Springs. Cowles Distinguished Professor Emeritus Department of Mathematics Trinity University San Antonio, Texas, USA [email protected] Differential equations are fundamental to many fields, with applications such as describing spring-mass systems and circuits and modeling control systems. This course presents techniques for solving and approximating solutions to ordinary differential equations. Definitions and Terminology. In Problems 35–38 ﬁnd an explicit solution of the given initial-value problem. Calculus Video Playlist:. This allows us to solve separable differential equations more conveniently, as demonstrated in the example below. 5 9/04/2008 Mixing Problems & Linear Equations 2. It shows that the non-linear fourth-order equation for the stream function with three independent variables admits functional separable solutions described by a system of three partial differential. Example Find the general solution of the differential equation Example Find the general solution of the differential equation. FIRST-ORDER DIFFERENTIAL EQUATIONS. Robotics/Motion Control/Mechatronics. Setting up mixing problems as separable differential equations. Each Differential Equations problem is tagged down to the core, underlying concept that is being tested. Separable equations have the form dy/dx = f(x) g(y), and are called separable because the variables x and y can be brought to opposite sides of the equation. We will now look at some examples of solving separable differential equations. Solution: The diﬀerential equation is separable. 7 Constant solutions In general, a solution to a diﬀerential equation is a function. 4 Separable Equations and Applications 30 1. 3 First-Order Separable Diﬀerential Equations 3 1. An equation is defined as separable if simple algebra operations can obtain a result such as the one discussed above (putting distinct variables in the equation apart in each side of the. A calculator for solving differential equations. Problems and Solutions for Ordinary Di ferential Equations by Willi-Hans Steeb International School for Scienti c Computing at University of Johannesburg, South Africa and by Yorick Hardy Department of Mathematical Sciences at University of South Africa, South Africa updated: February 8, 2017. 3 Slope Fields and Solution Curves 17 1. Definitions and Terminology. Linear Equations - Identifying and solving linear first order differential equations. 4 Direction Fields 5 1. As a rule of thumb, it requires nconditions to determine values for all narbitrary constants in the general solution of an nth-order differential equation (one condition for each. Looks like ‘homogeneous but for constant’ but is ‘almost separable’ 3. Ordinary differential equations: first-order linear equations, separable equations, exact equations, second-order linear equations, nonhomogeneous equations, systems of first-order linear equations, systems of nonlinear equations, modeling and applications. We show two methods that can be used to solve the given separable differential equation. These equations will be called later separable equations. We will look at some techniques that can be used. 6) is similar by a little more complex than that for the homogeneous equation in (3. General Solution of a Differential Equation. Linear Equations. Linear Equations - Identifying and solving linear first order differential equations. The order of the differential equation is the order of the highest order derivative present in the equation. The distinction between the singular and the general solution is just an algebraic distinction. 3, instead of using the solution formulas, as in Example1. Equilibrium Solutions. When an applied problem leads to a differential equation, there are usually conditions in the problem that determine speciﬁc values for the arbitrary constants. Identify separable simple differential equations. So the final solution of. The method of separation of variables is applied to the population growth in Italy and to an example of water leaking from a cylinder. And that diverges to minus infinity when x approaches 1. 18 Problems: Heat Equation 255 5. Homogenous Equations: is homogeneous if the function f(x,y) is homogeneous, that is By substitution, we consider the new function The new differential equation satisfied by z is which is a separable equation. 6) makes the DE non-homogeneous The solution of ODE in Equation (3. Exact Differential Equations and Integrating Factors. Solution Curves Without a Solution. How to solve exponential growth and decay word problems. Solve separable di erential equations and initial value problems. We look at a separable differential equation which, I am sure you recall, looks something like , then write it using Leibniz notation (which is more precise, at the expense of being somewhat intimidating… a topic for another time!) as. Solution Curves Without a Solution. Mixing Salt & Water with Separable Differential Equations Please give me an Upvote and Resteem if you have found this tutorial helpful. Free PDF download of NCERT Solutions for Class 12 Maths Chapter 9 - Differential Equations solved by Expert Teachers as per NCERT (CBSE) Book guidelines. Initial-Value Problems 1. Differential equations are fundamental to many fields, with applications such as describing spring-mass systems and circuits and modeling control systems. To find linear differential equations solution, we have to derive the general form or representation of the solution. (b) Find the particular solution yfx= ( ) to the differential equation with the initial condition f (−11)= and state its domain. The following video provides an outline of all the topics you would expect to see in a typical Differential Equations class (i. How to solve first order differential equations. Differential Equations and Linear Superposition • Basic Idea: Provide solution in closed form • Like Integration, no general solutions in closed form •Order of equation: highest derivative in equation e. First-Order Differential Equations 1 1. Ordinary differential equations We work with real numbers in this worksheet. Differential equations arise in a situation when we understand how various factors cause a quantity to change. First-Order Diﬀerential Equations and Their Applications 1 1. Separable differential equation Why is there. Solutions to differential equations involve the answer in the form of y = f(x). It is still easy to check that an implicit solution satisfies the differential equation. How to solve first order differential equations. 1: Solution Curves Without a Solution ; 2. Paradigm Find all the solutions to $\displaystyle\frac{dy}{dx}=xy+\frac yx$. Most of the time the independent variable is dropped from the writing and so a diﬀerential equation as (1. Afterwards, we will find the general solution and use the initial condition to find the particular solution. As a rule of thumb, it requires nconditions to determine values for all narbitrary constants in the general solution of an nth-order differential equation (one condition for each. Hence the derivatives are partial derivatives with respect to the various variables. Contents: How to solve separable differential equations - Separable differential equations - How to solve initial value problems-Linear - first-order differential equations - First order, linear differential equation - Linear differential equations, first order - Homogeneous first order ordinary differential equation - How to solve ANY differential equation - Mixing problems and. 4 Linear Equations 54 2. Definition. They’re word problems that require us to create a separable differential equation based on the concentration of a substance in a tank. This is called an initial-valued problem (IVP). First Order Equations Separable Ry0(x) = f(x)g(y) dy is a solution of the homogeneous equation, is a solution of the homogeneous problem. Often a radical change in the form of the solution of a differen- tial equation corresponds to a very small change in either the initial condition or the equation itself. Order of a differential equation is defined as the order of the highest order derivative of the dependent variable with respect to the independent variable involved in the given differential equation. Modeling Examples. Use the reduction of order to find a second solution. Lesson Summary. 4 Separable Differential Equations ¶ A differential equation is an equation for an unknown function that involves the derivative of the unknown function. Lec8: Partial derivatives and integration. Step 4: The initial condition means when. A(x) dx + B(y) dy = 0, where A(x) is a function of x only and B(y) is a function of y only. These revision exercises will help you practise the procedures involved in solving differential equations. Mixing Salt & Water with Separable Differential Equations Please give me an Upvote and Resteem if you have found this tutorial helpful. Looks like ‘homogeneous but for constant’ but is ‘almost separable’ 3. If you're behind a web filter, please make sure that the domains *. 6) makes the DE non-homogeneous The solution of ODE in Equation (3. 2: Separable. Solution of First Order Linear Differential Equations Linear and non-linear differential equations A differential equation is a linear differential equation if it is expressible in the form Thus, if a differential equation when expressed in the form of a polynomial involves the derivatives and dependent variable in the first power and there are no product […]. 2 Power Series, Analytic Functions, and the Taylor Series Method 431 8. Differential Equations as Mathematical Models. Differential equations are equations that have x, y, and the derivative of y with respect to x. In this section, we will try to apply differential equations to real life situations. Dependence of Solutions on Initial Conditions. Separable Variables. We will examine the role of complex numbers and how useful they are in the study of ordinary differential equations in a later chapter, but for the moment complex numbers will just muddy the situation. Initial Value Problems Example equation of the solution curve. Nonlinear DE: More complicated functions of y and its derivatives appear as well as multiplication by a constant or a function of x. Following example is the equation 1. This method is only possible if we can write the differential equation in the form. First Order Differential Equations. Solutions by Substitutions. Use * for multiplication a^2 is a 2.