2.100   ODE No. 100

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

$$a+x{y}^{\prime }(x)+xy(x{)}^2=0$$ ✓ Mathematica : cpu = 0.00914194 (sec), leaf count = 133

$$\left\{\{y(x)\to \frac{i\sqrt{-a}(c_1-2)\sqrt{x}{J}_0\left(2i\sqrt{-a}\sqrt{x}\right)+c_1(J_1\left(2i\sqrt{-a}\sqrt{x}\right)-i\sqrt{-a}\sqrt{x}{J}_2\left(2i\sqrt{-a}\sqrt{x}\right))}{2(c_1-1)x{J}_1\left(2i\sqrt{-a}\sqrt{x}\right)}\}\right\}$$

✓ Maple : cpu = 0.085 (sec), leaf count = 59

$$\{y\left(x\right)=1\sqrt{a}({\mathit{J}}_0\left(2\sqrt{a}\sqrt{x}\right)\mathit{\_}\mathit{C}\mathit{1}+{\mathit{Y}}_0\left(2\sqrt{a}\sqrt{x}\right))\frac{1}{\sqrt{x}}{(\mathit{\_}\mathit{C}\mathit{1}{\mathit{J}}_1\left(2\sqrt{a}\sqrt{x}\right)+{\mathit{Y}}_1\left(2\sqrt{a}\sqrt{x}\right))}^{-1}\}$$

Hand solution

$$\begin{array}{rl}x{y}^{\prime }+x{y}^2+a& =0\\ y^{\prime }& =-\frac{a}{x}-y^2\end{array}$$

This is Riccati first order non-linear. Let $y=-\frac{u^{\prime }}{uR}=\frac{u^{\prime }}{u}$, hence $y^{\prime }=\frac{u^{\prime \prime }}{u}-\frac{{\left(u^{\prime }\right)}^2}{u^2}$. Equating this to RHS of the above gives$$\begin{array}{rl}\frac{u^{\prime \prime }}{u}-\frac{{\left(u^{\prime }\right)}^2}{u^2}& =-\frac{a}{x}-{\left(\frac{u^{\prime }}{u}\right)}^2\\ \frac{u^{\prime \prime }}{u}& =-\frac{a}{x}\\ u^{\prime \prime }+\frac{a}{x}u& =0\end{array}$$

This is linear second order, an Emden Fowler ODE, with removal singularity. Solved using power series method. The solution is$$u=C_1\sqrt{x}\mathrm{BesselJ}(1,2\sqrt{ax})+C_2\sqrt{x}\mathrm{BesselY}(1,2\sqrt{ax})$$ But $$\frac{d}{dx}\mathrm{BesselJ}(1,2\sqrt{ax})=\frac{\sqrt{a}}{\sqrt{x}}(\mathrm{BesselJ}(0,2\sqrt{ax})-\frac{1}{2}\frac{1}{\sqrt{ax}}\mathrm{BesselJ}(1,2\sqrt{ax}))$$ And$$\frac{d}{dx}\mathrm{BesselY}(1,2\sqrt{ax})=\frac{\sqrt{a}}{\sqrt{x}}(\mathrm{BesselY}(0,2\sqrt{ax})-\frac{1}{2}\frac{1}{\sqrt{ax}}\mathrm{BesselY}(1,2\sqrt{ax}))$$ Therefore, $$\begin{array}{rl}{u}^{\prime }& =C_1(\frac{1}{2\sqrt{x}}\mathrm{BesselJ}(1,2\sqrt{a}\sqrt{x})+\sqrt{x}\frac{\sqrt{a}}{\sqrt{x}}(\mathrm{BesselJ}(0,2\sqrt{a}\sqrt{x})-\frac{1}{2}\frac{1}{\sqrt{ax}}\mathrm{BesselJ}(1,2\sqrt{a}\sqrt{x})))\\ & +C_2(\frac{1}{2\sqrt{x}}\mathrm{BesselY}(1,2\sqrt{a}\sqrt{x})+\sqrt{x}\frac{\sqrt{a}}{\sqrt{x}}(\mathrm{BesselY}(0,2\sqrt{ax})-\frac{1}{2}\frac{1}{\sqrt{ax}}\mathrm{BesselY}(1,2\sqrt{a}\sqrt{x})))\end{array}$$

Which is simplified to$$u^{\prime }=C_1\sqrt{a}\mathrm{BesselJ}(0,2\sqrt{a}\sqrt{x})+C_2\sqrt{a}\mathrm{BesselY}(0,2\sqrt{a}\sqrt{x})$$ Therefore, from $y=\frac{u^{\prime }}{u}$, the solution is$$y=\frac{C_1\sqrt{a}\mathrm{BesselJ}(0,2\sqrt{a}\sqrt{x})+C_2\sqrt{a}\mathrm{BesselY}(0,2\sqrt{a}\sqrt{x})}{C_1\sqrt{x}\mathrm{BesselJ}(1,2\sqrt{a}\sqrt{x})+C_2\sqrt{x}\mathrm{BesselY}(1,2\sqrt{a}\sqrt{x})}$$ Let $C=\frac{C_1}{C_2}$, hence$$y=\frac{C\sqrt{a}\mathrm{BesselJ}(0,2\sqrt{a}\sqrt{x})+\sqrt{a}\mathrm{BesselY}(0,2\sqrt{a}\sqrt{x})}{C\sqrt{x}\mathrm{BesselJ}(1,2\sqrt{a}\sqrt{x})+\sqrt{x}\mathrm{BesselY}(1,2\sqrt{a}\sqrt{x})}$$ Verification

restart; 
ode:=x*diff(y(x),x)+x*y(x)^2+a=0; 
num:=_C1*sqrt(a)*BesselJ(0,2*sqrt(a)*sqrt(x))+sqrt(a)*BesselY(0,2*sqrt(a)*sqrt(x)); 
den:=_C1*sqrt(x)*BesselJ(1,2*sqrt(a)*sqrt(x))+sqrt(x)*BesselY(1,2*sqrt(a)*sqrt(x)); 
my_solution:=num/den; 
odetest(y(x)=my_solution,ode); 
0