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CBSE Class 12 · 2012 · Differential EquationsLong Answer

Solve the following differential equation:

2 x^{2} \frac{d y}{d x} - 2 x y + y^{2} = 0

Answer: $\frac{2 x}{y} = lo g ∣ x ∣ + C$

Solution

\frac{d y}{d x} = \frac{2 x y - y ^{2}}{2 x ^{2}}

is homogeneous. Put

y = v x

:

v + x \frac{d v}{d x} = v - \frac{v ^{2}}{2} \Rightarrow x \frac{d v}{d x} = - \frac{v ^{2}}{2}

\int \frac{- 2 d v}{v ^{2}} = \int \frac{d x}{x} \Rightarrow \frac{2}{v} = lo g ∣ x ∣ + C

With

v = \frac{y}{x}

:

\frac{2 x}{y} = lo g ∣ x ∣ + C

CBSE Class 12 Board · 2012 · Subjective

Practise more: all CBSE board PYQs on Differential Equations →

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