Fill in the blanks. The solution of the differential equation x d… - Vidyadip

Question

Fill in the blanks.
The solution of the differential equation $\text{x}\frac{\text{dy}}{\text{dx}}+2\text{y}=\text{x}^2$ is ________.

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Answer

The solution of the differential equation $\text{x}\frac{\text{dy}}{\text{dx}}+2\text{y}=\text{x}^2$ is $\text{y}=\frac{\text{x}^4}{4}+\text{C}\text{x}^{-2}.$Solution:
We have, $\text{x}\frac{\text{dy}}{\text{dx}}+2\text{y}=\text{x}^2$
$\Rightarrow\frac{\text{dy}}{\text{dx}}+\frac{2\text{y}}{\text{x}}=\text{x}$
This equation the form $\frac{\text{dy}}{\text{dx}}+\text{Py}=\text{Q}$
$\therefore\text{I.F.}=\text{e}^{\int\frac{2}{\text{x}}\text{dx}}$
$=\text{e}^{2\log\text{x}}=\text{x}^2$
The general solution is
$\text{y}\text{x}^2=\int\text{x}.\text{x}^2\text{dx}+\text{C}$
$\Rightarrow\text{y}^{\text{x}^2}=\frac{\text{x}^4}{4}+\text{C}$
$\Rightarrow\text{y}=\frac{\text{x}^4}{4}+\text{C}\text{x}^{-2}$

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