2.29   ODE No. 29

1. Problem in Latex
2. Mathematica input
3. Maple input

$$y^{\prime }(x)-xy(x{)}^2-3xy(x)=0$$ ✓ Mathematica : cpu = 0.0239502 (sec), leaf count = 39

$$\left\{\{y(x)\to -\frac{3e^{3c_1+\frac{3x^2}{2}}}{e^{3c_1+\frac{3x^2}{2}}-1}\}\right\}$$

✓ Maple : cpu = 0.015 (sec), leaf count = 19

$$\{y\left(x\right)=3{(-1+3{\mathrm{e}}^{-3/2x^2}\mathit{\_}\mathit{C}\mathit{1})}^{-1}\}$$

Hand solution

$$\begin{array}{rl}{y}^{\prime }-x{y}^2-3xy& =0\\ y^{\prime }& =3xy+x{y}^2\\ \text{(1)}& & =P\left(x\right)+Q\left(x\right)y+R\left(x\right)y^2\end{array}$$

This is Bernoulli first order non-linear ODE since $P\left(x\right)=0$. To solve Bernoulli we always start by dividing by $y^2$$$\frac{y^{\prime }}{y^2}=\frac{3x}{y}+x$$ Then we let $u=\frac{1}{y}$, hence $u^{\prime }=\frac{-y^{\prime }}{y^2}$, therefore the above becomes$$\begin{array}{rl}-u^{\prime }& =3xu+x\\ u^{\prime }+3ux& =-x\end{array}$$

Integrating factor is $e^{\int 3xdx}=e^{\frac{3x^2}{2}}$, hence $$d\left(e^{\frac{3x^2}{2}}u\right)=-x{e}^{\frac{3x^2}{2}}$$ Integrating both sides gives$$\begin{array}{rl}{e}^{\frac{3x^2}{2}}u& =\int -x{e}^{\frac{3x^2}{2}}dx+C\\ & =-\frac{1}{3}{e}^{\frac{3x^2}{2}}+C\end{array}$$

Hence from above$$u=e^{\frac{-3x^2}{2}}(-\frac{1}{3}{e}^{\frac{3x^2}{2}}+C)$$

And since $y=\frac{1}{u}$ then$$y=\frac{e^{\frac{3x^2}{2}}}{C-\frac{1}{3}{e}^{\frac{3x^2}{2}}}$$ Verification

eq:=diff(y(x),x)-x*y(x)^2-3*x*y(x) = 0; 
sol:=exp(3*x^2/2)/(_C1- 1/3*exp(3*x^2/2)); 
odetest(y(x)=sol,eq); 
0