The differential equation x dy dx -y=x^ 2 has the general solutio… - Vidyadip

MCQ

The differential equation $\text{x}\frac{\text{dy}}{\text{dx}}-\text{y}=\text{x}^{2}$ has the general solution:

A
$y-x^3=2 C x$
✓
$2 y-x^3=C x$
C
$2 y+x^2=2 C x$
D
$y+x^2=2 C x$

✓

Answer

Correct option: B.

$2 y-x^3=C x$

We have,
$\text{x}\frac{\text{dy}}{\text{dx}}-\text{y}=\text{x}^{2}$
$\Rightarrow \frac{\text{dy}}{\text{dx}}-\frac{1}{\text{x}}\text{y}=\text{x}^{2}$
Comparing with we get,
$\text{P}=-\frac{1}{\text{x}}$
$\text{Q}=\text{x}^{2}$
Now,
$\text{I.F}=\text{e}^{-\int\frac{1}{\text{x}}\text{dx}}$
$=\text{e}^{-\log|\text{x}|}$
$=\text{e}^{\log|\frac{1}{\text{x}}|}$
$=\frac{1}{\text{x}}$
$\text{y}\times\text{I.F}=\int\text{x}^{2}\times\text{I.F}\text{dx}+\text{C}$
$\Rightarrow \text{y}\frac{1}{\text{x}}=\int\text{x}^{2}\times\frac{1}{\text{x}}\text{dx}+\text{C}$
$\Rightarrow \text{y}\frac{1}{\text{x}}=\int\text{x}^{2}\text{dx}+\text{C}$
$\Rightarrow \text{y}\frac{1}{\text{x}}=\frac{\text{x}^{2}}{2}+\text{C}$
$\Rightarrow 2\text{y}-\text{x}^{3}=\text{Cx}$

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