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CBSE Class 12 · 2011 · Differential EquationsLong Answer

Solve the following differential equation:

x d y - y d x = x^{2} + y^{2} d x

Answer: $y + x^{2} + y^{2} = C x^{2}$

Solution

Rearrange:

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

(homogeneous). Put

y = v x

:

v + x \frac{d v}{d x} = v + 1 + v^{2} \Rightarrow \frac{d v}{1 + v ^{2}} = \frac{d x}{x}

lo g v + 1 + v^{2} = lo g ∣ x ∣ + lo g C \Rightarrow v + 1 + v^{2} = C x

With

v = \frac{y}{x}

:

y + x^{2} + y^{2} = C x^{2}

CBSE Class 12 Board · 2011 · Subjective

Practise more: all CBSE board PYQs on Differential Equations →

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