# Integrating Factor: DE Question

For this question:

Can someone please explain the solution below, especially why the integrating factor is $e^{3x/2}$:

• Do you know what the integrating factor is and how to find it? – Kaster Feb 19 '15 at 18:22
• @Kaster No not really – Bailee Tucker Feb 19 '15 at 18:24
• Here you go - Integrating factor. – Kaster Feb 19 '15 at 18:27

The first step should be obvious, since the initial value problem states that $y(0)=4$, then in some neighbourhood of $x=0$ we introduce $$w(x)=y^{3/2}(x),$$ which allows to write the differential equation as $$\frac 23 w'(x)+w(x)=1,\quad w(0)=8.$$ This falls into a quite general category of linear ODEs with constant coefficients: say, we want to solve $$z'(x)+az(x)=b(x),$$ then we can multiply both parts of this equation by $e^{ xa}$ in order to obtain $$(e^{ xa}z(x))'= e^{xa}b(x).$$