Choose the correct answer from the given four options.The general… - Vidyadip

MCQ

Choose the correct answer from the given four options.The general solution of the differential equation $\frac{\text{dy}}{\text{dx}}\ \text{e}^{\frac{\text{x}^2}{2}}+\text{xy}$ is:

A
$\text{y}=\text{c}\text{e}^{\frac{-\text{x}^2}{2}}$
B
$\text{y}=\text{c}\text{e}^{\frac{\text{x}^2}{2}}$
✓
$\text{y}=(\text{x}+\text{c})\text{e}^{\frac{\text{x}^2}{2}}$
D
$\text{y}=(\text{c}-\text{x})\text{e}^{\frac{\text{x}^2}{2}}$

✓

Answer

Correct option: C.

$\text{y}=(\text{x}+\text{c})\text{e}^{\frac{\text{x}^2}{2}}$

Given that, $\frac{\text{dy}}{\text{dx}}\ \text{e}^{\frac{\text{x}^2}{2}}+\text{xy}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}-\text{xy}=\text{e}^{\frac{\text{x}^2}{2}}$
It is a linear differential equation.
Here, $\text{P}=-\text{x},\text{ Q}=\text{e}^{\frac{\text{x}^2}{2}}$
$\therefore\text{I.F.}=\text{e}^{\int-\text{xdx}}=\text{e}^{\frac{-\text{x}^2}{2}}$
The general solution is
$\text{y}\text{e}^{\frac{-\text{x}^2}{2}}=\int\text{e}^{\frac{-\text{x}^2}{2}}.\text{e}^{\frac{-\text{x}^2}{2}}\text{dx}+\text{c}$
$\Rightarrow\text{y}{\text{e}}^{\frac{-\text{x}^2}{2}}=\int1\text{dx}+\text{c}$
$\Rightarrow\text{y}\text{e}^{\frac{-\text{x}^2}{2}}=\text{x}+\text{c}$
$\Rightarrow\text{y}=\text{x}\text{e}^{\frac{\text{x}^2}{2}}+\text{c}\text{e}^{\frac{\text{x}^2}{2}}$
$\Rightarrow\text{y}=(\text{x}+\text{c})\text{e}^{\frac{\text{x}^2}{2}}$

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