Solve the following differential equation:
$2 x^{2} \frac{dy}{dx} - 2 x y + y^{2} = 0$

Solution

$Here, 2 x^{2} \frac{dy}{dx} - 2 x y + y^{2} = 0 \frac{dy}{dx} = \frac{2 x y - y^{2}}{2 x^{2}}$

Hence the given equation is an homogeneous equation.

Let y = v x

$and \frac{dy}{dx} = v + x \frac{dv}{dx} So, v + x \frac{dv}{dx} = \frac{2 x (v x) - (v x)^{2}}{2 x^{2}} = \frac{2 v - v^{2}}{2} = v - \frac{v^{2}}{2} \Rightarrow x \frac{dv}{dx} = - \frac{v^{2}}{2} \Rightarrow 2 \int \frac{1}{v^{2}} dv = - \int \frac{dx}{x} \Rightarrow 2 (- \frac{1}{v}) = - \log |x| + C \Rightarrow \frac{2 x}{y} = \log |x| + C$

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Vector Algebra

Solve the following differential equation:
$2 x^{2} \frac{dy}{dx} - 2 x y + y^{2} = 0$

Some More Questions From Vector Algebra Chapter

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NCERT Sample Papers

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For Daily Free Study Material Join wiredfaculty

Vector Algebra

Solve the following differential equation:2 x2 dydx - 2 x y + y2 = 0

Some More Questions From Vector Algebra Chapter

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Solve the following differential equation:
$2 x^{2} \frac{dy}{dx} - 2 x y + y^{2} = 0$