Differential equation

Question

Solve the first order linear differential equation

dy/dx + 2y = x-7. y(0)=0

Answer 1

The integrating factor being μ (x) = e^∫2dx = e^2x

we have

e^2x [ dy/dx + 2y] = e^2x [x-7]

e^2x dy + 2y e^2xdx = [x e^2x -7 e^2x] d x

d [ e^2x y] = [x e^2x -7 e^2x] d x

∫ d [ e^2x y] = ∫ [x e^2x -7 e^2x] d x

e^2x y = ∫ x e^2x d x - ∫ 7 e^2x d x

e^2x y = (1/2) ∫ x de^2x - ∫ 7 e^2x d x =

e^2x y = (1/2) x e^2x - (1/4) e^2x - (7/2)e^2x + ξ , where ξ any real number

y = (1/2) x - (1/4) - (7/2) + ξ e^-2x

y = (1/2) x - (15/4) + ξ e^-2x but y(0)=0 ⇒ 0 = - (15/4) + ξ

y = (1/2) x - 15/4 + (15/4) e^-2x