61.29.38 problem 147

Internal problem ID [12568]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-4
Problem number : 147
Date solved : Thursday, March 13, 2025 at 11:43:49 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x \left (2 a \,x^{n}+b \right ) y^{\prime }+\left (a^{2} x^{2 n}+a \left (b +n -1\right ) x^{n}+\alpha \,x^{2 m}+\beta \,x^{m}+\gamma \right ) y&=0 \end{align*}

✓ Maple. Time used: 0.608 (sec). Leaf size: 111

ode:=x^2*diff(diff(y(x),x),x)+x*(2*a*x^n+b)*diff(y(x),x)+(a^2*x^(2*n)+a*(b+n-1)*x^n+alpha*x^(2*m)+beta*x^m+gamma)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = x^{-\frac {b}{2}} x^{-\frac {m}{2}} \sqrt {x}\, {\mathrm e}^{-\frac {a \,x^{n}}{n}} \left (c_{1} \operatorname {WhittakerM}\left (-\frac {i \beta }{2 m \sqrt {\alpha }}, \frac {\sqrt {b^{2}-2 b -4 \gamma +1}}{2 m}, \frac {2 i \sqrt {\alpha }\, x^{m}}{m}\right )+c_{2} \operatorname {WhittakerW}\left (-\frac {i \beta }{2 m \sqrt {\alpha }}, \frac {\sqrt {b^{2}-2 b -4 \gamma +1}}{2 m}, \frac {2 i \sqrt {\alpha }\, x^{m}}{m}\right )\right ) \]

✓ Mathematica. Time used: 0.791 (sec). Leaf size: 291

ode=x^2*D[y[x],{x,2}]+x*(2*a*x^n+b)*D[y[x],x]+(a^2*x^(2*n)+a*(b+n-1)*x^n+\[Alpha]*x^(2*m)+\[Beta]*x^m+\[Gamma])*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to x^{\frac {1}{2}-\frac {m}{2}} 2^{\frac {1}{2} \left (\frac {\sqrt {m^2 \left (b^2-2 b-4 \gamma +1\right )}}{m^2}+1\right )} \left (x^n\right )^{-\frac {b}{2 n}} \left (x^m\right )^{\frac {1}{2} \left (\frac {\sqrt {m^2 \left (b^2-2 b-4 \gamma +1\right )}}{m^2}+1\right )} e^{-\frac {a x^n}{n}+\frac {i \sqrt {\alpha } x^m}{m}} \left (c_1 \operatorname {HypergeometricU}\left (\frac {m^2-\frac {i \beta m}{\sqrt {\alpha }}+\sqrt {m^2 \left (b^2-2 b-4 \gamma +1\right )}}{2 m^2},\frac {m^2+\sqrt {m^2 \left (b^2-2 b-4 \gamma +1\right )}}{m^2},-\frac {2 i x^m \sqrt {\alpha }}{m}\right )+c_2 L_{-\frac {m^2-\frac {i \beta m}{\sqrt {\alpha }}+\sqrt {m^2 \left (b^2-2 b-4 \gamma +1\right )}}{2 m^2}}^{\frac {\sqrt {m^2 \left (b^2-2 b-4 \gamma +1\right )}}{m^2}}\left (-\frac {2 i x^m \sqrt {\alpha }}{m}\right )\right ) \]

✗ Sympy

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

ValueError : Expected Expr or iterable but got None