2.109   ODE No. 109

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

$$x{y}^{\prime }(x)-y(x)(2y(x)\log (x)-1)=0$$ ✓ Mathematica : cpu = 0.0693812 (sec), leaf count = 17

$$\left\{\{y(x)\to \frac{1}{2\log (x)+c_{1}x+2}\}\right\}$$ ✓ Maple : cpu = 0.012 (sec), leaf count = 15

$$\{y\left(x\right)={(2+\mathit{\_}\mathit{C}\mathit{1}x+2\ln \left(x\right))}^{-1}\}$$

Hand solution

$x{y}^{\prime }+ax{y}^2+2y+bx=0$This is Riccati non-linear first order. Converting it to standard form$$\begin{array}{rl}x{y}^{\prime }-y(2y\ln x-1)& =0\\ x{y}^{\prime }& =y(2y\ln x-1)\\ \text{(1)}& y^{\prime }& =-\frac{1}{x}y+y^2\frac{2}{x}\ln x{y}^{\prime }& =f_0+f_{1}y+f_2{y}^2\end{array}$$

This is Bernoulli non-linear first order ODE since $f_0=0$. Dividing by $y^2$ gives$$\frac{y^{\prime }}{y^2}=-\frac{1}{x}\frac{1}{y}+\frac{2}{x}\ln x$$ Putting $u=\frac{1}{y}$, hence $u^{\prime }=-\frac{y^{\prime }}{y^2}$, and the above becomes$$\begin{array}{rl}-u^{\prime }& =-\frac{1}{x}u+2\frac{\ln x}{x}\\ -u^{\prime }+\frac{1}{x}u& =2\frac{\ln x}{x}\\ u^{\prime }-\frac{1}{x}u& =-2\frac{\ln x}{x}\end{array}$$

Integrating factor is $\mu =e^{\int -\frac{1}{x}dx}=e^{-\ln x}=\frac{1}{x}$, hence$$\begin{array}{rl}d\left(\mu u\right)& =-2\mu \frac{\ln x}{x}\\ d\left(\frac{1}{x}u\right)& =-2\frac{\ln x}{x^2}\end{array}$$

Integrating$$\begin{array}{rl}\frac{1}{x}u& =-2\int \frac{1}{x^2}\ln xdx+C\\ & =-2(-\frac{\ln x}{x}-\frac{1}{x})+C\end{array}$$

Therefore$$\begin{array}{rl}u& =-2x(-\frac{\ln x}{x}-\frac{1}{x})+Cx\\ & =2(\ln x+1)+Cx\end{array}$$

Since $u=\frac{1}{y}$ then$$y=\frac{1}{2(\ln x+1)+Cx}$$ Verification

restart; 
ode:=x*diff(y(x),x)-y(x)*(2*y(x)*ln(x)-1)=0; 
my_solution:=1/(2*(ln(x)+1)+_C1*x); 
odetest(y(x)=my_solution,ode); 
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