Lineal diferencial equation?

Question

hi. how do i get to the answer using the integrating factor.

equation: dx/dy + y = e^3x

answer: y= 1/4 * e^3x + ce^-x

Answer 1

i have no idea what that means,

=l

Answer 2

i assume you meant dy/dx instead of dx/dy

y' + y = e^(3x)

integrating factor:

I(x) = e^(∫1dx) = e^x

multiply e^x for both sides

e^x * y' + e^x * y = e^x  * e^(3x)

reverse product rule on the left side:

d/dx (y e^x) = e^(4x)

integrate both sides

y e^x = ∫e^(4x) dx

y e^x  = (1/4)e^(4x) + C

y = (1/4)e^(3x) + C/e^x

y = (1/4) e^(3x) + Ce^(-x)