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Differential Equations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations5 Marks
Question
Solve the following differential equation:
$\text{y e}^{\frac{\text{x}}{\text{y}}}\text{dx}=\big(\text{xe}^{\frac{\text{x}}{\text{y}}}+\text{y}\big)\text{dy}$
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Answer

We have,
$\text{y e}^{\frac{\text{x}}{\text{y}}}\text{dx}=\big(\text{xe}^{\frac{\text{x}}{\text{y}}}+\text{y}\big)\text{dy}$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\frac{\text{xe}^{\frac{\text{x}}{\text{y}}}+\text{y}}{\text{y e}^{\frac{\text{x}}{\text{y}}}}$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\frac{\frac{\text{x}}{\text{y}}\text{e}^{\frac{\text{x}}{\text{y}}}+1}{\text{e}^{\frac{\text{x}}{\text{y}}}}$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\frac{\text{x}}{\text{y}}+\text{e}^{\frac{-\text{x}}{\text{y}}}$
This is a homogeneous differential equation.
Putting y = vx and $\frac{\text{dy}}{\text{dx}}=\text{v + y}\frac{\text{dv}}{\text{dx}}$, we get
$\text{v + y}\frac{\text{dv}}{\text{dx}}=\text{v + e}^{-\text{v}}$
$\Rightarrow\ \text{y}\frac{\text{dv}}{\text{dy}}=\text{e}^{-\text{v}}$
$\Rightarrow\ \text{e}^{\text{v}}\text{dv}=\frac{1}{\text{y}}\text{dy}$
Integrating both sides, we get
$\int\text{e}^{\text{v}}\text{dv}=\int\frac{1}{\text{y}}\text{dy}$
$\Rightarrow\ \text{e}^{\text{v}}=\log|\text{y}|+\text{C}$
Putting $\text{v}=\frac{\text{y}}{\text{x}}$, we get
$\Rightarrow\ \text{e}^{\frac{\text{x}}{\text{y}}}=\log|\text{y}|+\text{C}$
Hence, $\text{e}^{\frac{\text{x}}{\text{y}}}=\log|\text{y}|+\text{C}$ is the required solution.

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