Solve the following initial value problem dydx 6y 10x 3 y

Solve the following initial value problem dy/dx + 6y = 10x + 3 y(0) = 9

Solution

Ans-

Find all solutions to dy dx = 4x + 5y 5x + 4y . This is a homogeneous equation; we substitute y = xv to obtain v + x dv dx = 4 + 5v 5 + 4v ; x dv dx = 4 4v 2 5 + 4v . This last equation is separable; we note the singular solutions v = ±1 and separate the variables: 5 + 4v 1 v 2 dv = 4 dx x . Integrating, we obtain Z 5 + 4v 1 v 2 dv = 4 ln |x| + C. To calculate the integral on the left hand side, we use the method of elementary fractions. We write 5 + 4v 1 v 2 = 5 + 4v (1 v)(1 + v) = A 1 v + B 1 + v , where A and B are constants. Then A(1 + v) + B(1 v) = 5 + 4v; A B = 4 A + B = 5 = A = 9 2 B = 1 2 ; Z 5 + 4v 1 v 2 dv = 1 2 Z 9 1 v + 1 1 + v dv = 1 2 ln 1 + v (1 v) 9 + C. Thus 1 2 ln 1 + v (1 v) 9 = 4 ln |x| + C. Exponentiating both sides and taking into account the singular solutions, we conclude that 1 + v (1 v) 9 = Cx8 or v = 1. It only remains to substitute v = y/x and simplify. Answer: x + y (x y) 9 = C or y = x