Given:
[tex]\frac{dy}{dx}=2yx+yx^2[/tex]
Find: the exact solution y=f(x) to the differential equation, f(0)=1 and domain of the function.
Explanation:
[tex]\begin{gathered} dy=(2yx+yx^2)dx \\ dy=y(2x+x^2)dx \\ \frac{dy}{y}=2xdx+x^2dx \\ \end{gathered}[/tex]
on integrating both side,
[tex]logy=x^2+\frac{x^3}{3}+c[/tex]
put f(0)=1,
[tex]\begin{gathered} log1=0^2+\frac{0^3}{3}+c \\ c=0 \end{gathered}[/tex][tex]logy=x^2+\frac{x^3}{3}[/tex]
