Solve the following differential equation: x dy – y dx = x^ 2 +y^… - Vidyadip

Question

Solve the following differential equation:
x dy – y dx = $\sqrt{\text{x}^{2}+\text{y}^{2}}\text{ dx}$ .

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Answer

Given equation can be written as $\frac{\text{dy}}{\text{dx}}=\frac{\text{y}}{\text{x}}+\sqrt{1+\Big(\frac{\text{y}}{\text{x}}\Big)^{2}}$
$\Rightarrow\text{v + x }\frac{\text{dv}}{\text{dx}}=\text{v}+\sqrt{1+\text{v}^{2}}$ where $\frac{\text{y}}{\text{x}}=\text{v}$
$\Rightarrow\int\frac{\text{dv}}{\sqrt{1 + \text{v}^{2}}}=\int\frac{\text{dx}}{\text{x}}$
$\Rightarrow\log|\text{v}+\sqrt{1+\text{v}^{2}}|=\log\text{cx}$
$\Rightarrow\text{v}+\sqrt{1+\text{v}^{2}}=\text{cx}\therefore\text{y}+\sqrt{\text{x}^{2}+\text{y}^{2}}=\text{cx}^{2}$.

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