55.2.6 problem 6

Internal problem ID [13232]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. 1.2.2. Equations Containing Power Functions
Problem number : 6
Date solved : Wednesday, October 01, 2025 at 03:42:37 AM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 343

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

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

✓ Mathematica. Time used: 0.732 (sec). Leaf size: 764

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

\begin{align*} y(x)&\to -\frac {x^n \left (\sqrt {b} c_1 (n+1) \sqrt {-(n+1)^2} \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {a} c}{\sqrt {b} \sqrt {-(n+1)^2}}+\frac {n}{n+1}\right ),\frac {n}{n+1},\frac {2 \sqrt {a} \sqrt {b} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+c_1 \left (\sqrt {a} c (n+1)+\sqrt {b} \sqrt {-(n+1)^2} n\right ) \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {a} c}{\sqrt {b} \sqrt {-(n+1)^2}}+\frac {3 n+2}{n+1}\right ),\frac {n}{n+1}+1,\frac {2 \sqrt {a} \sqrt {b} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+\sqrt {b} (n+1) \sqrt {-(n+1)^2} \left (L_{-\frac {\sqrt {a} c}{2 \sqrt {b} \sqrt {-(n+1)^2}}-\frac {n}{2 (n+1)}}^{-\frac {1}{n+1}}\left (\frac {2 \sqrt {a} \sqrt {b} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+2 L_{-\frac {\sqrt {a} c}{2 \sqrt {b} \sqrt {-(n+1)^2}}-\frac {3 n+2}{2 n+2}}^{\frac {n}{n+1}}\left (\frac {2 \sqrt {a} \sqrt {b} x^{n+1}}{\sqrt {-(n+1)^2}}\right )\right )\right )}{\sqrt {a} (n+1)^2 \left (L_{-\frac {\sqrt {a} c}{2 \sqrt {b} \sqrt {-(n+1)^2}}-\frac {n}{2 (n+1)}}^{-\frac {1}{n+1}}\left (\frac {2 \sqrt {a} \sqrt {b} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+c_1 \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {a} c}{\sqrt {b} \sqrt {-(n+1)^2}}+\frac {n}{n+1}\right ),\frac {n}{n+1},\frac {2 \sqrt {a} \sqrt {b} x^{n+1}}{\sqrt {-(n+1)^2}}\right )\right )}\\ y(x)&\to \frac {x^n \left (-\frac {\left (\sqrt {a} c (n+1)+\sqrt {b} \sqrt {-(n+1)^2} n\right ) \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {a} c}{\sqrt {b} \sqrt {-(n+1)^2}}+\frac {n}{n+1}+2\right ),\frac {n}{n+1}+1,\frac {2 \sqrt {a} \sqrt {b} x^{n+1}}{\sqrt {-(n+1)^2}}\right )}{\operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {a} c}{\sqrt {b} \sqrt {-(n+1)^2}}+\frac {n}{n+1}\right ),\frac {n}{n+1},\frac {2 \sqrt {a} \sqrt {b} x^{n+1}}{\sqrt {-(n+1)^2}}\right )}-\sqrt {b} \sqrt {-(n+1)^2} (n+1)\right )}{\sqrt {a} (n+1)^2} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
n = symbols("n") 
y = Function("y") 
ode = Eq(-a*y(x)**2 - b*x**(2*n) - c*x**(n - 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*n) - c*x**(n - 1) + Derivative(y(x), x) cannot be solved by the lie group method