1.3.4.1.3 Example \(y^{\prime }=5xe^{3x}+3y+10xe^{-3x}y^{2}\) Comparing to \(y^{\prime }=f_{0}+f_{1}y+f_{2}y^{2}\) shows that

\begin{align*} f_{0} & =5xe^{3x}\\ f_{1} & =3\\ f_{2} & =10xe^{-3x}\end{align*}

Let

\begin{align*} y & =u\left ( x\right ) e^{\phi }\\ \phi & =\int f_{1}dx \end{align*}

Therefore \(\phi =\int 3dx=3x\) and we have

\[ u^{\prime }=F\left ( x\right ) +G\left ( x\right ) u^{2}\]

Where

\begin{align*} F\left ( x\right ) & =f_{0}e^{-\phi }=5xe^{3x}e^{-3x}=5x\\ G\left ( x\right ) & =f_{2}e^{\phi }=10xe^{-3x}e^{3x}=10x \end{align*}

Hence

\[ u^{\prime }=5x+10xu^{2}\]

Since \(F\left ( x\right ) \) is proportional to \(G\left ( x\right ) \), then this is separable ode which is easily solved. Once \(u\) is known, then \(y\) is found since \(y=u\left ( x\right ) e^{\phi }\).