Question

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

$2x^{4}ydy+y^{2}dy+4x^3{y}^{2}dx-x^{2}dx=0$
$2x^{3}y\left(xdy+2ydx\right)+y^{2}dy-x^{2}dx=0$
$2x^{2}y\left(x^{2}dy+2xydx\right)+y^{2}dy-x^{2}dx=0$
$2\left(x^{2}y\right)d\left(x^{2}y\right)+y^{2}dy-x^{2}dx=0$
On integrating, we get,
${\left(x^{2}y\right)}^2+\frac{y^3}{3}-\frac{x^3}{3}=C_1$
$3x^4{y}^2+y^3-x^3=C$