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 : Tuesday, January 28, 2025 at 03:16:38 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.052 (sec). Leaf size: 31

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

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

✓ Solution by Mathematica

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

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

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