49.15.12 problem 8

Internal problem ID [7698]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 3. Linear equations with variable coefficients. Page 130
Problem number : 8
Date solved : Sunday, March 30, 2025 at 12:19:12 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-2 x y^{\prime }+2 \alpha y&=0 \end{align*}

✓ Maple. Time used: 0.033 (sec). Leaf size: 31

ode:=diff(diff(y(x),x),x)-2*x*diff(y(x),x)+2*alpha*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = x \left (\operatorname {KummerM}\left (\frac {1}{2}-\frac {\alpha }{2}, \frac {3}{2}, x^{2}\right ) c_1 +\operatorname {KummerU}\left (\frac {1}{2}-\frac {\alpha }{2}, \frac {3}{2}, x^{2}\right ) c_2 \right ) \]

✓ Mathematica. Time used: 0.036 (sec). Leaf size: 45

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

\[ y(x)\to c_1 \cosh \left (\alpha \log \left (\sqrt {x^2-1}+x\right )\right )+i c_2 \sinh \left (\alpha \log \left (\sqrt {x^2-1}+x\right )\right ) \]

✗ Sympy

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

False