60.3.147 problem 1151

Internal problem ID [11157]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1151
Date solved : Monday, January 27, 2025 at 10:47:14 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 42

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

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

✓ Solution by Mathematica

Time used: 0.088 (sec). Leaf size: 88

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

\[ y(x)\to -\frac {i \sqrt {\frac {2}{\pi }} \sqrt {x} \left (\left (i \sqrt {a} c_2 x+c_1\right ) \sinh \left (\sqrt {a} x\right )-\left (\sqrt {a} c_1 x+i c_2\right ) \cosh \left (\sqrt {a} x\right )\right )}{\left (-i \sqrt {a} x\right )^{3/2}} \]