5.7 Differential Equations: Separation of Variables Calculus
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1 5.7 DIFFERENTIAL EQUATIONS: SEPARATION OF VARIABLES In the last section we discussed the method of separation of variables to solve a differential equation. In this section we have 4 basic goals. (1) Verif solutions to differential equations () Solving differential equations using the separation of variables technique with or without initial values given (3) Recognizing Homogeneous Functions (4) Solving Homogeneous Differential Equations through changes of variables Verif Solutions To verif a solution to a differential equation, we need to show that the equation is satisfied when the given function and its derivatives are substituted into the equation. Example: Determine whether or not each of the given functions is a solution to the differential equation '' = 0. (a) = sin x (b) = 4e x Solving Differential Equations Due to the fact ou must integrate to solve a differential equation, the solutions to differential equations contain constants. If our final answer includes these constants, this is a general solution. However, if we are given some initial conditions, then we are able to solve for those constants. A solution which is obtained from these initial conditions is called a particular solution. Example: Suppose our general solution to a differential equation was to this general solution. = Cx. Sketch the famil of solution curves relating x Example: Suppose that ou were given the initial condition that the when x = 1, = 3. Label this point and sketch the particular solution on the same grid above. Can ou determine the value of C? 1
2 Example: Find the general solution of ( x ) + 4 d = x dx Example: Given the initial condition ( 0) =1, find the particular solution of the equation x x dx + e 1 d = 0 13
3 Homogeneous Differential Equations There are certain equations that cannot be separated. For instance, tr solving the equation ( x ) dx+ 3x d=0 using the separation of variables technique. What problems do ou encounter? One of the tpes of equations where this occurs is differential equations of the form ' = f ( x, ), where f is a homogeneous function. Using this notation, the derivative of is a function of both x and. Definition: A function f is a homogeneous function of degree n (n is a real number) provided that n f tx, t = t f x, ( Example: Determine whether or not the following functions are homogeneous. 3 (a) f x, = x 4x + 3x ) (b) f x, = x+ Definition: A homogeneous differential equation is an equation of the form M x, dx+ N x, d= 0 where M and N are homogeneous functions of the same degree. To solve a homogeneous differential equation b the method of separation of variables, we use the following change of variables theoerem. Theorem 5.17 Change of Variables for Homogeneous Equations If M x, dx+ N x, d= 0 is homogeneous, then it can be transformed into a differential equation whose variables are separable b the substitution where v is a differentiable function of x. = vx 14
4 Example: Find the general solution of ( x ) dx+ 3x d= 0. 15
5 General Guidelines for Solving a Homogeneous Differential Equation n 1. Recognize that our equation is an homogeneous equation; that is, ou need to check that f tx, t = t f x,.. Make the substitution = xv. (Remember to find d = x dv + v dx) 3. Solve the new equation b separation of variables x and v. (If this is not possible, then either the equation was not homogeneous or ou made a mistake!) 4. Substitute v = back into the answer. x 5. If initial conditions are given, solve for the constant(s). Example: Solve 3 3 x + ' = x 16
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