Question

The solution of the differential equation $y (2x^{4}+y)\frac{dy}{dx}=(1-4xy^{2})x^{2}$ is given by

• A. $3(x^{2}y)^{2}+y^{3}-x^{3}=c$
• B. $xy^{2}+\frac{y^{3}}{3}-\frac{x^{3}}{3}+c=0$
• C. $\frac{2}{5}yx^{5}+\frac{y^{3}}{3}=\frac{x^{3}}{3}-\frac{4xy^{3}}{3}+c$
• D. None of these

Answer 1

Answer: (A)

$y (2x^{4}+y)\frac{dy}{dx}=(1-4xy^{2})x^{2}$
or $2x^{4}y\,dy+y^{2}\,dy+4x^{3}y^{2}dx-x^{2}\,dx=0$
or $2x^{2}y (x^{2}\,dy +2xy\,dx)+y^{2}\,dy-x^{2}\,dx=0$
or $2x^{2}\,y \,d(x^{2}y)+y^{2}\,dy-x^{2}\,dx=0$
Integrating, we get $(x^{2}y)^{2}+\frac{y^{3}}{3}-\frac{x^{3}}{3}=c$
or $3(x^{2}y)^{2}+y^{3}-x^{3}=c$