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Question

# In this exercise, you will need to solve the differential equation given below.$2 y^{\prime}-y=x e^{x / 2}$

Solution

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Step 1
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Note that the standard form of a first order differential equation is

$\frac{dy}{dx}+a(x)y=b(x).$

We can rewrite the differential equation in standard form.

\begin{aligned}2\frac{dy}{dx}-y&=xe^{x/2}\\ \frac{dy}{dx}-\dfrac{y}{2}&=\dfrac{x}{2}e^{x/2}.&&\textcolor{#4257b2}{\text{(Divide both sides by 2)}}\\ \end{aligned}

where $a(x)=-\frac{1}{2},b(x)=\frac{x}{2}e^{x/2}$.

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