The solution of the differential equation x y² dy - (x³ + y³) dx = 0 is

Correct Answer is : y3=3x3log⁡(cx)y³ = 3 x³ (c x)y3=3x3log(cx)

Correct Answer is : y3=3x3log⁡(cx)y³ = 3 x³ (c x)y3=3x3log(cx)

Test Index

EAMCET Engineering 2008 Solved Paper

Section: Mathematics

Question : 160 of 160

Marks: +1, -0

x y^2 dy - (x^3 + y^3) dx = 0

Solution:

Given differential equation can be rewritten as

\frac{dy}{dx} = \frac{x^3 + y^3}{x y^2}

It is a homogeneous differential equation.

\text{Put } y = v x \Rightarrow \frac{dy}{dx} = v + x \frac{dv}{dx}

\therefore x \frac{dv}{dx} + v = \frac{x^3 + v^3 x^3}{x^3 v^2}

\Rightarrow x \frac{dv}{dx} + v = \frac{1+v^3}{v^2}

\Rightarrow x \frac{dv}{dx} = \frac{1}{v^2}

\Rightarrow v^2 dv = \frac{dx}{x}

On integrating both sides, we get

\frac{v^3}{3} = \log x + \log c

\Rightarrow \frac{1}{3} \left(\frac{y}{x}\right)^3 = \log x + \log c

\Rightarrow y^3 = 3 x^3 \log c x

Go to Question:

Prev Question