Question

The solution of differential equation $\left(2y+xy^{3}\right)dx+\left(x+x^{2}y^{2} \right)dy=0$ is

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

Answer 1

Answer: (A)

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