60.3.220 problem 1224

Internal problem ID [11230]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1224
Date solved : Monday, January 27, 2025 at 10:49:23 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} \left (x^{2}+1\right ) y^{\prime \prime }+x y^{\prime }+a y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 23

dsolve((x^2+1)*diff(diff(y(x),x),x)+x*diff(y(x),x)+a*y(x)=0,y(x), singsol=all)
 

\[ y = c_{1} \sin \left (\sqrt {a}\, \operatorname {arcsinh}\left (x \right )\right )+c_{2} \cos \left (\sqrt {a}\, \operatorname {arcsinh}\left (x \right )\right ) \]

✓ Solution by Mathematica

Time used: 0.041 (sec). Leaf size: 30

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

\[ y(x)\to c_1 \cos \left (\sqrt {a} \text {arcsinh}(x)\right )+c_2 \sin \left (\sqrt {a} \text {arcsinh}(x)\right ) \]