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CBSE Class 12 · 2022 · Differential EquationsLong Answer

Find the general solution of the differential equation

(x^{3} + y^{3}) d y = x^{2} y d x

.

Answer: $lo g ∣ y ∣ - \frac{x ^{3}}{3 y ^{3}} = C$

Solution

\frac{d y}{d x} = \frac{x ^{2} y}{x ^{3} + y ^{3}}

is homogeneous. Put

y = v x

:

v + x \frac{d v}{d x} = \frac{v}{1 + v ^{3}} \Rightarrow x \frac{d v}{d x} = \frac{- v ^{4}}{1 + v ^{3}}

\int \frac{1 + v ^{3}}{v ^{4}} d v = - \int \frac{d x}{x} \Rightarrow - \frac{1}{3 v ^{3}} + lo g ∣ v ∣ = - lo g ∣ x ∣ + C

With

v = \frac{y}{x}

(so

lo g v + lo g x = lo g y

):

lo g ∣ y ∣ - \frac{x ^{3}}{3 y ^{3}} = C

CBSE Class 12 Board · 2022 · Subjective

Practise more: all CBSE board PYQs on Differential Equations →

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