Question

Solve the following differential equations :
$\sqrt{(a+x)}\frac{dy}{dx}+xy=0$

Answer 1

$\sqrt{(a+n)}\frac{dy}{dx}+xy=0$

$\frac{dy}{dx}=\frac{-xy}{\sqrt{a+x}}$

$\frac{dy}{y}=\frac{-xdx}{\sqrt{a+x}}$

integrate both sides,

$|n|y|+|n|c|=\int \frac{-xdx}{\sqrt{a+x}}$

$a+x=t$

$x=(t-a)$

$dx=dt$

In $|cy|=-\int \frac{(t-a)dt}{t^2}$

In $|cy|=-|n|t|=\frac{-a}{t}$

$t_{cy}=e^{-a/t}$

$\Rightarrow ey/(x=a)=e^{a/(x+a)}$

$\Rightarrow y=k(x+a)e^{a/(x+a)}$