54.1.30 problem 30

Internal problem ID [11344]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 30
Date solved : Tuesday, September 30, 2025 at 07:57:27 PM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }+x^{-a -1} y^{2}-x^{a}&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 51

ode:=diff(y(x),x)+x^(-a-1)*y(x)^2-x^a = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {x^{\frac {1}{2}+a} \left (-\operatorname {BesselK}\left (a +1, 2 \sqrt {x}\right ) c_1 +\operatorname {BesselI}\left (a +1, 2 \sqrt {x}\right )\right )}{\operatorname {BesselK}\left (a , 2 \sqrt {x}\right ) c_1 +\operatorname {BesselI}\left (a , 2 \sqrt {x}\right )} \]

✓ Mathematica. Time used: 0.251 (sec). Leaf size: 265

ode=D[y[x],x] + x^(-a-1)*y[x]^2 - x^a==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {x^a \left (\sqrt {x} \operatorname {Gamma}(1-a) \operatorname {BesselI}\left (-a-1,2 \sqrt {x}\right )+\sqrt {x} \operatorname {Gamma}(1-a) \operatorname {BesselI}\left (1-a,2 \sqrt {x}\right )-a \operatorname {Gamma}(1-a) \operatorname {BesselI}\left (-a,2 \sqrt {x}\right )+(-1)^a c_1 \sqrt {x} \operatorname {Gamma}(a+1) \operatorname {BesselI}\left (a-1,2 \sqrt {x}\right )-(-1)^a a c_1 \operatorname {Gamma}(a+1) \operatorname {BesselI}\left (a,2 \sqrt {x}\right )+(-1)^a c_1 \sqrt {x} \operatorname {Gamma}(a+1) \operatorname {BesselI}\left (a+1,2 \sqrt {x}\right )\right )}{2 \left (\operatorname {Gamma}(1-a) \operatorname {BesselI}\left (-a,2 \sqrt {x}\right )+(-1)^a c_1 \operatorname {Gamma}(a+1) \operatorname {BesselI}\left (a,2 \sqrt {x}\right )\right )}\\ y(x)&\to \frac {x^a \left (\sqrt {x} \operatorname {BesselI}\left (a-1,2 \sqrt {x}\right )-a \operatorname {BesselI}\left (a,2 \sqrt {x}\right )+\sqrt {x} \operatorname {BesselI}\left (a+1,2 \sqrt {x}\right )\right )}{2 \operatorname {BesselI}\left (a,2 \sqrt {x}\right )} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-x**a + x**(-a - 1)*y(x)**2 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -x**a + x**(-a - 1)*y(x)**2 + Derivative(y(x), x) cannot be solved by the lie group method