60.1.14 problem 14

Internal problem ID [10028]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 14
Date solved : Sunday, March 30, 2025 at 02:54:09 PM
CAS classification : [[_Riccati, _special]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 187

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

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

✓ Mathematica. Time used: 0.392 (sec). Leaf size: 639

ode=D[y[x],x] + y[x]^2 + a*x^m==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") 
m = symbols("m") 
y = Function("y") 
ode = Eq(a*x**m + y(x)**2 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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