Determine the integrating factor for solving the DE (simplify your answer, but do NOT solve the DE): (x^2 + 5x)y' - xy = 3x - 1

solution

(x^2+5x)y'-xy=3x-1

comapre with  linear first order o.d.e.

dy /dx + P(x)y = Q(x)

p(x)= -\frac{x}{x^2+5x}= -\frac{1}{x+5}

IF= e^{\int p(x)}dx

IF= e^{\int -\frac{1}{x+5}}dx

IF= e^{-ln|x+5|}

IF= \frac{1}{x+5}

IF= e R P (x)dx