60.3.154 problem 1158

Internal problem ID [11164]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1158
Date solved : Monday, January 27, 2025 at 10:47:29 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+a y^{\prime }-\left (b^{2} x^{2}+a b \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.788 (sec). Leaf size: 178

dsolve(x^2*diff(diff(y(x),x),x)+a*diff(y(x),x)-(b^2*x^2+a*b)*y(x)=0,y(x), singsol=all)
 

\[ y = \sqrt {x}\, \left (\operatorname {HeunD}\left (4 \sqrt {2}\, \sqrt {a b}, -1-4 \sqrt {2}\, \sqrt {a b}, 8 \sqrt {2}\, \sqrt {a b}, -4 \sqrt {2}\, \sqrt {a b}+1, \frac {\sqrt {2}\, \sqrt {a b}\, x -a}{\sqrt {2}\, \sqrt {a b}\, x +a}\right ) {\mathrm e}^{\frac {-b \,x^{2}+a}{x}} c_{1} +\operatorname {HeunD}\left (-4 \sqrt {2}\, \sqrt {a b}, -1-4 \sqrt {2}\, \sqrt {a b}, 8 \sqrt {2}\, \sqrt {a b}, -4 \sqrt {2}\, \sqrt {a b}+1, \frac {\sqrt {2}\, \sqrt {a b}\, x -a}{\sqrt {2}\, \sqrt {a b}\, x +a}\right ) {\mathrm e}^{b x} c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.516 (sec). Leaf size: 38

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

\[ y(x)\to e^{b x} \left (c_2 \int _1^xe^{\frac {a}{K[1]}-2 b K[1]}dK[1]+c_1\right ) \]