Solve the following differential equation a+x dy+x dx=0 - Vidyadip

Question

Solve the following differential equation
$\sqrt{\text{a}+\text{x}}\text{dy}+\text{x dx}=0$

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Answer

We have,
$\sqrt{\text{a}+\text{x}}\text{dy}+\text{x dx}=0$
$\Rightarrow\sqrt{\text{a}+\text{x dy}}=-\text{x dx}$
$\Rightarrow\text{dy}=\frac{-\text{x}}{\sqrt{\text{a}+\text{x}}}\ \text{dx}$
$\Rightarrow\text{dy}=-\frac{(\text{x}+\text{a}-\text{a})}{\sqrt{\text{a}+\text{x}}}\ \text{dx}$
$\Rightarrow\text{dy}=-\Big(\sqrt{\text{a}+\text{x}}-\frac{\text{a}}{\sqrt{\text{a}+\text{x}}}\Big)\text{dx}$
Integrating both sides, we get
$\int\text{dy}=\int\Big(\sqrt{\text{a}+\text{x}}-\frac{\text{a}}{\sqrt{\text{a}+\text{x}}}\Big)\text{dx}$
$\Rightarrow\text{y}=-\frac{2(\text{a}+\text{x})^{\frac{3}{2}}}{3}+2\text{a}\sqrt{\text{a}+\text{x}}+\text{C}$
$\Rightarrow\text{y}+\frac{2}{3}(\text{a}+\text{x})^{\frac{3}{2}}-2\text{a}\sqrt{\text{a}+\text{x}}=\text{C}$
hence, $\text{y}+\frac{2}{3}(\text{a}+\text{x})^{\frac{3}{2}}-2\text{a}\sqrt{\text{a}+\text{x}}=\text{C}$ is the solution to the given differential equation.

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