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CBSE Class 12 · 2006 · Differential EquationsLong Answer

Solve the following differential equation:

2 x y d x + (x^{2} + 2 y^{2}) d y = 0

.

Answer: $3 x^{2} y + 2 y^{3} = C$

Solution

Here

M = 2 x y

and

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

. Since

\frac{\partial M}{\partial y} = 2 x = \frac{\partial N}{\partial x},

the equation is exact. Seek

F (x, y) = C

with

\frac{\partial F}{\partial x} = M

:

F = \int 2 x y d x = x^{2} y + g (y)

Then

\frac{\partial F}{\partial y} = x^{2} + g^{'} (y) = x^{2} + 2 y^{2}

gives

g^{'} (y) = 2 y^{2}

, so

g (y) = \frac{2 y ^{3}}{3}

. Thus

x^{2} y + \frac{2 y ^{3}}{3} = C_{1} \Rightarrow 3 x^{2} y + 2 y^{3} = C

CBSE Class 12 Board · 2006 · Subjective

Practise more: all CBSE board PYQs on Differential Equations →

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