The general solution of the differntial equation dy dx = y x is: - Vidyadip

MCQ

The general solution of the differntial equation $\frac{\text{dy}}{\text{dx}}=\frac{\text{y}}{\text{x}}$ is:

A
$\log\text{y}=\text{kx}$
✓
$\text{y}=\text{kx}$
C
$\text{xy}=\text{k}$
D
$\text{y}=\text{k}\log\text{x}$

✓

Answer

Correct option: B.

$\text{y}=\text{kx}$

We have,
$\frac{\text{dy}}{\text{dx}}=\frac{\text{y}}{\text{x}}$
$\Rightarrow\frac{\text{1}}{\text{y}}\text{dy}=\frac{\text{1}}{\text{x}}\text{dx}$
Integrating both sides, we get
$\int\frac{\text{1}}{\text{y}}\text{dy}=\int\frac{\text{1}}{\text{x}}\text{dx}$
$\log\text{y}=\log\text{x}+\log\text{k}$
$\log\text{y}-\log\text{x}=\log\text{k}$
$\log\frac{\text{y}}{\text{x}}=\log\text{k}$
$\Rightarrow\frac{\text{y}}{\text{x}}=\text{k}$
$\Rightarrow\text{y}=\text{k}{\text{x}}$

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