2.104   ODE No. 104

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

$$axy(x{)}^2+bx+x{y}^{\prime }(x)+2y(x)=0$$ ✓ Mathematica : cpu = 0.111434 (sec), leaf count = 43

$$\left\{\{y(x)\to -\frac{1}{ax}-\sqrt{\frac{b}{a}}\tan (ax\sqrt{\frac{b}{a}}-c_1)\}\right\}$$ ✓ Maple : cpu = 0.046 (sec), leaf count = 63

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

Hand solution

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

Using transformation suggested by Kamke $y=u\left(x\right)-\frac{1}{ax}$ then $y^{\prime }=u^{\prime }+\frac{1}{a{x}^2}$. Equating this to RHS of (1) gives$$\begin{array}{rl}{u}^{\prime }+\frac{1}{a{x}^2}& =-b-\frac{2}{x}(u-\frac{1}{ax})-a{(u-\frac{1}{ax})}^2\\ & =-b-\frac{2}{x}u+\frac{2}{a{x}^2}-a(u^2+\frac{1}{a^2{x}^2}-\frac{2u}{ax})\\ & =-b-\frac{2}{x}u+\frac{2}{a{x}^2}-a{u}^2-\frac{1}{a{x}^2}+\frac{2u}{x}\end{array}$$

Hence

$$\begin{array}{rl}{u}^{\prime }& =-b-a{u}^2\\ \frac{du}{dx}& =-b-a{u}^2\end{array}$$

This is separable

$$\frac{du}{b+a{u}^2}=-dx$$

Integrating

$$\begin{array}{rl}\int \frac{du}{b+a{u}^2}& =-x+C\\ \frac{1}{\sqrt{ba}}\arctan \left(\frac{au}{\sqrt{ba}}\right)& =-x+C\\ \arctan \left(\frac{au}{\sqrt{ba}}\right)& =-\sqrt{ba}x+C\\ u& =\frac{\sqrt{ba}}{a}\tan (-\sqrt{ba}x+C)\end{array}$$

Hence

$$\begin{array}{rl}y& =u-\frac{1}{ax}\\ & =\frac{\sqrt{ba}}{a}\tan (-\sqrt{ba}x+C)-\frac{1}{ax}\end{array}$$

Verification

restart; 
ode:=x*diff(y(x),x)+a*x*y(x)^2+2*y(x)+b*x = 0; 
my_solution:=sqrt(b*a)/a*tan(-sqrt(b*a)*x+_C1)-1/(a*x); 
odetest(y(x)=my_solution,ode); 
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