Solution of differential equation 2xy dy dx = x^2 + 3 y^2 is (whe… - Vidyadip

MCQ

Solution of differential equation $2xy\frac{{dy}}{{dx}} = {x^2} + 3{y^2}$ is

(where $p$ is a constant)

A
${x^3} + {y^2} = p{x^2}$
B
$\frac{{{x^2}}}{2} + \frac{{{y^3}}}{x} = {y^2} + p$
C
${x^2} + {y^3} = p{x^2}$
✓
${x^2} + {y^2} = p{x^3}$

✓

Answer

Correct option: D.

${x^2} + {y^2} = p{x^3}$

d
(d) It is homogeneous equation $\frac{{dy}}{{dx}} = \frac{{{x^2} + 3{y^2}}}{{2xy}}$

Put $y = vx$ and $\frac{{dy}}{{dx}} = v + x\frac{{dv}}{{dx}}$

So, we get $x\frac{{dv}}{{dx}} = \frac{{1 + {v^2}}}{{2v}}$

==> $\frac{{2vdv}}{{1 + {v^2}}} = \frac{{dx}}{x}$

On integrating, we get ${x^2} + {y^2} = p{x^3}$.

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