Question

The equation of the curve passing through the origin and satisfying the equation $(1+x^2)\frac{dy}{dx}+2xy=4x^2$ is

• A. $3(1+x^2)y=4x^3$
• B. $3(1-x^2)y=4x^3$
• C. $3(1+x^2)=x^3$
• D. None of these

Answer 1

The correct option is A

$3(1+x^2)y=4x^3$

$\frac{dy}{dx}+\frac{2x}{1+x^2}y=\frac{4x^2}{1+x^2}$
It is linear equation of the form $\frac{dy}{dx}+Py=Q$
Here $P=\frac{2x}{1+x^2}andQ=\frac{4x^2}{1+x^2}$
$I.F.e^{\int \frac{2x}{1+x^2}dx}=e^{log(1+x^2)}=(1+x^2)$
Therefore, solution is given by
$y.(1+x^2)=\int \frac{4x^2}{1+x^2}(1+x^2)dx+c$
$=\frac{4x^3}{3}+c.$
But it passes through (0,0) therefore c = 0,
Hence the curve is $3y(1+x^2)=4x^3.$