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For this question:

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

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  • $\begingroup$ Do you know what the integrating factor is and how to find it? $\endgroup$ – Kaster Feb 19 '15 at 18:22
  • $\begingroup$ @Kaster No not really $\endgroup$ – Bailee Tucker Feb 19 '15 at 18:24
  • $\begingroup$ Here you go - Integrating factor. $\endgroup$ – Kaster Feb 19 '15 at 18:27
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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 $$(e^{ xa}z(x))'= e^{xa}b(x).$$

Now on the left we have a derivative of the unknown function, and on the right - a know function, hence we are able to solve it.

In more general case, you need to study the characteristic polynomial of the linear ODE with constant coefficients. This wiki article should be a good start.

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