Question

The solution of the differential equation $y\left(2 x^{4} + y\right)dy+\left(4 x y^{2} - 1\right)x^{2}dx=0$ is (where $C$ is an arbitrary constant)

• A. $3x^{2}y+x^{3}-y^{3}=C$
• B. $3x^{4}y^{2}+y^{3}-x^{3}=C$
• C. $3x^{2}y^{4}+x^{3}-y^{3}=C$
• D. $3x^{2}y^{4}+y^{3}-x^{3}=C$

Answer 1

Answer: (B)

$2x^{4}ydy+y^{2}dy+4x^{3}y^{2}dx-x^{2}dx=0$
$2x^{3}y\left(x d y + 2 y d x\right)+y^{2}dy-x^{2}dx=0$
$2x^{2}y\left(x^{2} d y + 2 x y d x\right)+y^{2}dy-x^{2}dx=0$
$2\left(x^{2} y\right)d\left(x^{2} y\right)+y^{2}dy-x^{2}dx=0$
On integrating, we get,
$\left(x^{2} y\right)^{2}+\frac{y^{3}}{3}-\frac{x^{3}}{3}=C_{1}$
$3x^{4}y^{2}+y^{3}-x^{3}=C$