Differential Equations Question - CBSE

Question

Solve the homogeneous differential equation $x \dfrac{dy}{dx} = y - x 	an\left(\dfrac{y}{x}ight)$.

Accepted Answer

Substitute $y = vx \Rightarrow x \dfrac{dv}{dx} = - an v$. [1.5 Marks]
$\int \cot v dv = -\int \dfrac{dx}{x}$. [1.5 Marks]
$\log|\sin v| = \log\left|\dfrac{C}{x} ight| \Rightarrow \sin(y/x) = \dfrac{C}{x}$. [1.0 Mark]
$x \sin\left(\dfrac{y}{x} ight) = C$. [1.0 Mark]

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🎯 Official CBSE Marking Scheme:
Substituting $y = vx$: 1.5 Marks
Separating variables and integrating cotangent: 1.5 Marks
Eliminating logarithms: 1.0 Mark
Evaluating final solution $x \sin(y/x) = C$: 1.0 Mark

Question:

Step-by-Step Solution

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