Find the general solution?

Question

Find the general solution of dy/dx+3y=e^3x.

Answer 1

Use the integrating factor method.

μ dy/dx + 3μy = μ e^3x

Let μ satisfy dμ/dx = 3μ.

Substitution yields μ dy/dx + y dμ/dx = μe^3x.

The left side is a product rule for (d/dx) μy so you can integrate both sides.

μy = ∫ μe^3x dx

y = (1/μ) ∫ μe^3x dx

Now find μ from it's differential equation which is separable.

dμ/dx = 3μ

dμ/μ = 3 dx.

∫ dμ/μ = ∫ 3 dx

ln |μ| = 3x + C

μ = e^3x+C = e^ce^3x = Ae^3x

Plug in any specific μ, say μ = e^3x into the original solution of y.

y = e^-3x ∫ e^3xe^3x dx = e^-3x ∫ e^6xdx = e^-3x (e^6x/6 + C) = e^3x/6 + Ce^-3x

Check:

(d/dx) (e^3x/6 + Ce^-3x) + 3(e^3x/6 + Ce^-3x) = e^3x/2 - 3Ce^-3x + e^3x/2 + 3Ce^-3x = e^3x.