2.30   ODE No. 30

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

$$x^{-a-1}y(x{)}^2-x^a+y^{\prime }(x)=0$$ ✓ Mathematica : cpu = 0.0655547 (sec), leaf count = 230

$$\left\{\{y(x)\to \frac{x^{a+1}(c_1(\frac{1}{2}{x}^{-\frac{a}{2}-\frac{1}{2}}\mathrm{\Gamma }(a+1)(I_{a-1}\left(2\sqrt{x}\right)+I_{a+1}\left(2\sqrt{x}\right))-\frac{1}{2}a{x}^{-\frac{a}{2}-1}\mathrm{\Gamma }(a+1)I_a\left(2\sqrt{x}\right))-\frac{1}{2}(-1{)}^{-a}a{x}^{-\frac{a}{2}-1}\mathrm{\Gamma }(1-a)I_{-a}\left(2\sqrt{x}\right)+\frac{1}{2}(-1{)}^{-a}{x}^{-\frac{a}{2}-\frac{1}{2}}\mathrm{\Gamma }(1-a)(I_{-a-1}\left(2\sqrt{x}\right)+I_{1-a}\left(2\sqrt{x}\right)))}{c_1{x}^{-a/2}\mathrm{\Gamma }(a+1)I_a\left(2\sqrt{x}\right)+(-1{)}^{-a}{x}^{-a/2}\mathrm{\Gamma }(1-a)I_{-a}\left(2\sqrt{x}\right)}\}\right\}$$

✓ Maple : cpu = 0.085 (sec), leaf count = 54

$$\{y\left(x\right)=x^{a+1}(-{\mathit{K}}_{a+1}\left(2\sqrt{x}\right)\mathit{\_}\mathit{C}\mathit{1}+{\mathit{I}}_{a+1}\left(2\sqrt{x}\right))\frac{1}{\sqrt{x}}{({\mathit{K}}_a\left(2\sqrt{x}\right)\mathit{\_}\mathit{C}\mathit{1}+{\mathit{I}}_a\left(2\sqrt{x}\right))}^{-1}\}$$

Hand solution

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

This is Ricatti first order non-linear ODE. Using standard transformation$$y=-\frac{u^{\prime }}{uR\left(x\right)}=x^{a+1}\frac{u^{\prime }}{u}$$

Hence

$$y^{\prime }=(a+1)x^a\frac{u^{\prime }}{u}+x^{a+1}\frac{u^{\prime \prime }}{u}-x^{a+1}\frac{{\left(u^{\prime }\right)}^2}{u^2}$$

Comparing to (1) gives

$$\begin{array}{rl}{x}^a-x^{-a-1}{y}^2& =(a+1)x^a\frac{u^{\prime }}{u}+x^{a+1}\frac{u^{\prime \prime }}{u}-x^{a+1}\frac{{\left(u^{\prime }\right)}^2}{u^2}\\ x^a-x^{-a-1}{\left(x^{a+1}\frac{u^{\prime }}{u}\right)}^2& =(a+1)x^a\frac{u^{\prime }}{u}+x^{a+1}\frac{u^{\prime \prime }}{u}-x^{a+1}\frac{{\left(u^{\prime }\right)}^2}{u^2}\\ 1-\frac{x^{-a-1}}{x^a}{x}^{2a+2}\frac{{\left(u^{\prime }\right)}^2}{u^2}& =(a+1)\frac{u^{\prime }}{u}+x\frac{u^{\prime \prime }}{u}-x\frac{{\left(u^{\prime }\right)}^2}{u^2}\\ 1-x\frac{{\left(u^{\prime }\right)}^2}{u^2}& =(a+1)\frac{u^{\prime }}{u}+x\frac{u^{\prime \prime }}{u}-x\frac{{\left(u^{\prime }\right)}^2}{u^2}\\ 1& =(a+1)\frac{u^{\prime }}{u}+x\frac{u^{\prime \prime }}{u}\\ \text{(2)}& x{u}^{\prime \prime }+(1+a)u^{\prime }-u& =0\end{array}$$

This is Bessel like second order linear ODE. The solution is

$$u=C_1\frac{1}{\sqrt{x^a}}\mathrm{BesselI}(a,2\sqrt{x})+C_2\frac{1}{\sqrt{x^a}}\mathrm{BesselK}(a,2\sqrt{x})$$

But $$\begin{array}{rl}\frac{d}{dx}\frac{1}{\sqrt{x^a}}\mathrm{BesselI}(a,2\sqrt{x})& =\frac{1}{\sqrt{x^{1+a}}}\mathrm{BesselI}(1+a,2\sqrt{x})\\ \frac{d}{dx}\frac{1}{\sqrt{x^a}}\mathrm{BesselI}(a,2\sqrt{x})& =-\frac{1}{\sqrt{x^{1+a}}}\mathrm{BesselK}(1+a,2\sqrt{x})\end{array}$$

Hence

$$u^{\prime }=C_1\frac{1}{\sqrt{x^{1+a}}}\mathrm{BesselI}(1+a,2\sqrt{x})-C_2\frac{1}{\sqrt{x^{1+a}}}\mathrm{BesselK}(1+a,2\sqrt{x})$$

And from $y=x^{a+1}\frac{u^{\prime }}{u}$

$$y=x^{1+a}\frac{C_1\frac{1}{\sqrt{x^{1+a}}}\mathrm{BesselI}(1+a,2\sqrt{x})-C_2\frac{1}{\sqrt{x^{1+a}}}\mathrm{BesselK}(1+a,2\sqrt{x})}{C_1\frac{1}{\sqrt{x^a}}\mathrm{BesselI}(a,2\sqrt{x})+C_2\frac{1}{\sqrt{x^a}}\mathrm{BesselK}(a,2\sqrt{x})}$$

Let $C=\frac{C_2}{C_1}$ hence

$$y=x^{1+a}\frac{\frac{1}{\sqrt{x^{1+a}}}\mathrm{BesselI}(1+a,2\sqrt{x})-C\frac{1}{\sqrt{x^{1+a}}}\mathrm{BesselK}(1+a,2\sqrt{x})}{\frac{1}{\sqrt{x^a}}\mathrm{BesselI}(a,2\sqrt{x})+C\frac{1}{\sqrt{x^a}}\mathrm{BesselK}(a,2\sqrt{x})}$$

Or

$$\begin{array}{rl}y& =x^{1+a}\frac{x^{-\frac{1}{2}}\mathrm{BesselI}(1+a,2\sqrt{x})-C{x}^{-\frac{1}{2}}\mathrm{BesselK}(1+a,2\sqrt{x})}{\mathrm{BesselI}(a,2\sqrt{x})+C\mathrm{BesselK}(a,2\sqrt{x})}\\ & =\frac{x^{\frac{1}{2}+a}\mathrm{BesselI}(1+a,2\sqrt{x})-C{x}^{\frac{1}{2}+a}\mathrm{BesselK}(1+a,2\sqrt{x})}{\mathrm{BesselI}(a,2\sqrt{x})+C\mathrm{BesselK}(a,2\sqrt{x})}\end{array}$$

Verification

eq:=diff(y(x),x)+x^(-a-1)*y(x)^2-x^a = 0; 
num:=x^(1/2+a)*BesselI(1+a,2*sqrt(x))-_C1*x^(1/2+a)*BesselK(1+a,2*sqrt(x)); 
den:=BesselI(a,2*sqrt(x))+_C1*BesselK(a,2*sqrt(x)); 
my_sol:=num/den; 
odetest(y(x)=my_sol,eq); 
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