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Differential Equations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations1 Mark
MCQ
The general solution of differention eqution $\frac{\text{y}\ \text{dx}-\text{x}\ \text{dy}}{\text{y}}=0$ is :
  • A
    $xy = C$
  • B
    $x = Cy^2$
  • ✓
    $y = Cx$
  • D
    $y = Cx^2$
✓

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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