Applying two different methods, solve this DE: y' + (a + bx)y = f(x)

Answer 1

use an integrating factor

it's e^integral of (a+bx) = e^(ax+bx/2) = e^g(x)

y' + p(x)y = f(x) can be separable by multiplying by the integrating factor

e^g(x)y' + e^g(x)p(x)y = f(x)

d(e^g(x)y)/dx = f(x)

integrate both sides

e^g(x)y = F(x) + C

y = (F(x)+ C)/e^g(x)

y= (F(x)+C)e^(-ax-x^2/2) C is a constant, F(x) is the integral of f(x)

Answer 2

Multiplies both sides by integrating factor

Integral exp(ax+b) = (1/a)exp(ax+b).

It then becomes separable

Then undetermined coefficients or variation of parameters