54.1.25 problem 25

Internal problem ID [11339]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 25
Date solved : Tuesday, September 30, 2025 at 07:55:18 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 347

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

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

✓ Mathematica. Time used: 0.656 (sec). Leaf size: 928

ode=D[y[x],x] + a*y[x]^2 - b*x^(2*nu) - c*x^(nu-1)==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") 
c = symbols("c") 
nu = symbols("nu") 
y = Function("y") 
ode = Eq(a*y(x)**2 - b*x**(2*nu) - c*x**(nu - 1) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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