60.1.24 problem 24

Internal problem ID [10038]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 24
Date solved : Sunday, March 30, 2025 at 02:54:42 PM
CAS classification : [[_Riccati, _special]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 214

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

\[ y = \frac {-\sqrt {-a b}\, x^{\frac {\nu }{2}+1} \operatorname {BesselJ}\left (\frac {3+\nu }{\nu +2}, \frac {2 \sqrt {-a b}\, x^{\frac {\nu }{2}+1}}{\nu +2}\right ) c_1 -\operatorname {BesselY}\left (\frac {3+\nu }{\nu +2}, \frac {2 \sqrt {-a b}\, x^{\frac {\nu }{2}+1}}{\nu +2}\right ) \sqrt {-a b}\, x^{\frac {\nu }{2}+1}+c_1 \operatorname {BesselJ}\left (\frac {1}{\nu +2}, \frac {2 \sqrt {-a b}\, x^{\frac {\nu }{2}+1}}{\nu +2}\right )+\operatorname {BesselY}\left (\frac {1}{\nu +2}, \frac {2 \sqrt {-a b}\, x^{\frac {\nu }{2}+1}}{\nu +2}\right )}{x a \left (c_1 \operatorname {BesselJ}\left (\frac {1}{\nu +2}, \frac {2 \sqrt {-a b}\, x^{\frac {\nu }{2}+1}}{\nu +2}\right )+\operatorname {BesselY}\left (\frac {1}{\nu +2}, \frac {2 \sqrt {-a b}\, x^{\frac {\nu }{2}+1}}{\nu +2}\right )\right )} \]

✓ Mathematica. Time used: 0.408 (sec). Leaf size: 770

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

\begin{align*} \text {Solution too large to show}\end{align*}

✗ Sympy

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

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