y' = 4y + 5 + 5 e^(2x) Nonhomogeneous differential equation

Solve  y' = 4y + 5 + 5e^2x

An example of Nonhomogeneous differential equation

y' + P(x)y = Q(x) 형식으로 만들기

y' + (-4)y = 5e^2x

적분인자 찾기

M(x) = e^∫P(x)dx = e^∫(-4)dx = e^-4x

양변에 적분인자 e^-4x 곱하기

e^-4xy' - 4e^-4xy = e^-4x5e^2x = 5e^-2x

d/dx(e^-4xy) = 5e^-2x

양변 적분

∫d/dx(e^-4xy) dx = ∫5e^-2x dx

e^-4xy = (-5/2)e^-2x+c

y = e^4x(-5/2)e^-2x+c)

y = ce^4x - (5/2)e^2x