Question

Differential Equations problem

Answer 1

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Answer 2

multiply each side by \(\dfrac{1}{\sqrt{x}}\) to make it an exact DE

Answer 3

This is a 1st order linear DE.

\(\large\color{#000000 }{ \displaystyle y'-\frac{1}{2x}y=x}\)

\(\large\color{#000000 }{ \displaystyle y'e^{ \int dx/(2x)}-\frac{1}{2x}ye^{ \int dx/(2x)}=xe^{ \int dx/(2x)}}\)

and apply the product rule backwards,

Note that: \(\large\color{#000000 }{ \displaystyle e^{\ln(x)\cdot (1/2) }=\sqrt{e^{\ln(x)}} =\sqrt{x} }\)