Question

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

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

Answer 1

Answer: (A)

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