Question

In Problems 19-26, solve the given differential equation

$\mathbf{(}\mathbf{6}\mathit{x}\mathbf{+}\mathbf{1}\mathbf{)}{\mathit{y}}^{\mathbf{2}}\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}\mathbf{+}\mathbf{3}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{2}{\mathit{y}}^{\mathbf{3}}\mathbf{=}\mathbf{0}$

Short answer

The required answer is$x^3+2x{y}^3+\frac{y^3}{3}=c$

Step-by-step solution

(6x+1)y2dydx+3x2+2y3=0

Compare with $mdx+ndy=0$

m=$3x^2+2y^3$

n=$(6x+1)y^2$

 Now, differentiate m with respect to x and n with respect to y.

$$\begin{gathered} \frac{\left(deln\right)}{\left(delx\right)}=6y^2 \\ \frac{\left(delm\right)}{\left(dely\right)}=6y^2 \\ As\frac{delm}{dely}=\frac{deln}{delx} \end{gathered}$$

Therefore it is an exact differential equation

Final Answer

$\int (3x^2+2y^3)dx+\int y^{2}dy=c$

$x^3+2x{y}^3+\frac{y^3}{3}=c$is the required answer.