60.3.200 problem 1204

Internal problem ID [11210]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1204
Date solved : Monday, January 27, 2025 at 10:48:48 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 35

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

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

✓ Solution by Mathematica

Time used: 0.069 (sec). Leaf size: 84

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

\[ y(x)\to -\frac {a x^{3/2} e^{-\frac {1}{2} x (a+2 b)} \left (2 (i a c_2 x+2 c_1) \sinh \left (\frac {a x}{2}\right )-2 (a c_1 x+2 i c_2) \cosh \left (\frac {a x}{2}\right )\right )}{\sqrt {\pi } (-i a x)^{5/2}} \]