2.44   ODE No. 44

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

$$2a{x}^{3}y(x{)}^3+y^{\prime }(x)+2xy(x)=0$$ ✓ Mathematica : cpu = 0.0725068 (sec), leaf count = 72

$$\{\{y(x)\to -\frac{\sqrt{2}}{\sqrt{-2a{x}^2-a+2c_1{e}^{2x^2}}}\},\{y(x)\to \frac{\sqrt{2}}{\sqrt{-2a{x}^2-a+2c_1{e}^{2x^2}}}\}\}$$ ✓ Maple : cpu = 0.017 (sec), leaf count = 53

$$\{y\left(x\right)=-2\frac{1}{\sqrt{-4a{x}^2+4{\mathrm{e}}^{2x^2}\mathit{\_}\mathit{C}\mathit{1}-2a}},y\left(x\right)=2\frac{1}{\sqrt{-4a{x}^2+4{\mathrm{e}}^{2x^2}\mathit{\_}\mathit{C}\mathit{1}-2a}}\}$$

Hand solution

$$\begin{array}{}\text{(1)}& y^{\prime }=-2xy-2a{x}^3{y}^3\end{array}$$

This is of the form $y^{\prime }=f_0+f_{1}y+f_2{y}^2+f_3{y}^3$ where $f_0=0,f_2=0$. Hence this is Bernoulli first order non-linear ODE. We start by diving by $y^3$$$\frac{y^{\prime }}{y^3}=-2x\frac{1}{y^2}-2a{x}^3$$ Let $u=\frac{1}{y^2}$, hence $u^{\prime }=-2\frac{y^{\prime }}{y^3}$ and the above becomes$$\begin{array}{rl}-\frac{1}{2}{u}^{\prime }& =-2xu-2a{x}^3\\ u^{\prime }-4xu& =4a{x}^3\end{array}$$

Integrating factor is $e^{-4\int xdx}=e^{-2x^2}$ hence$$\frac{d}{dx}\left(e^{-2x^2}u\right)=4a{x}^3{e}^{-2x^2}$$ Integrating$$\begin{array}{rl}{e}^{-2x^2}u& =4a\int x^3{e}^{-2x^2}dx+C\\ & =4a\left(\frac{-1}{8}(2x^2+1)e^{-2x^2}\right)+C\end{array}$$

Therefore$$u=-\frac{1}{2}a(2x^2+1)+C{e}^{2x^2}$$ Hence$$y^2=\frac{1}{u}=\frac{1}{-\frac{1}{2}a(2x^2+1)+C{e}^{2x^2}}$$ Or$$y=\pm \frac{\sqrt{2}}{\sqrt{-a(2x^2+1)+C{e}^{2x^2}}}$$ Verification

ode:=2*a*x^3*y(x)^3+diff(y(x),x)+2*x*y(x)=0; 
my_sol:=sqrt(2)/sqrt(-a*(2*x^2+1)+_C1*exp(2*x^2)); 
odetest(y(x)=my_sol,ode); 
0 
my_sol:=-sqrt(2)/sqrt(-a*(2*x^2+1)+_C1*exp(2*x^2)); 
odetest(y(x)=my_sol,ode); 
0