The general solution of differention eqution y dx-x dy y =0 is: - Vidyadip

MCQ

The general solution of differention eqution $\frac{\text{y}\ \text{dx}-\text{x}\ \text{dy}}{\text{y}}=0$ is:

A
$xy = C$
B
$x=C y^2$
✓
$y = Cx$
D
$y=C x^2$

✓

Answer

Correct option: C.

$y = Cx$

We have,
$\frac{\text{y}\ \text{dx}-\text{x}\ \text{dy}}{\text{y}}=0$
$\Rightarrow \text{y}\ \text{dx}=\text{x}\ \text{dy}$
$\Rightarrow \frac{1}{\text{y}}\ \text{dy}=\frac{1}{\text{x}}\ \text{dx}$
Integrating both sides, we get,
$\int\frac{1}{\text{y}}\ \text{dy}=\int\frac{1}{\text{x}}\ \text{dx}$
$\Rightarrow \log\text{y}=\log\text{x}+\text{D}$
$\Rightarrow \log\text{y}-\log\text{x}=\text{C}$
$\Rightarrow \log\big(\frac{\text{y}}{2}\big)=\log\text{C}$
$\Rightarrow \frac{\text{y}}{2}=\text{C}$
$\Rightarrow \text{y}=\text{Cx}$

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