60.3.226 problem 1230

Internal problem ID [11236]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1230
Date solved : Monday, January 27, 2025 at 10:49:36 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

✓ Solution by Maple

Time used: 0.122 (sec). Leaf size: 36

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

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

✓ Solution by Mathematica

Time used: 0.047 (sec). Leaf size: 68

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

\[ y(x)\to \left (x^2+1\right )^{\frac {1}{2}-\frac {a}{4}} \left (c_1 P_{\frac {a-4}{2}}^{\frac {a-2}{2}}(i x)+c_2 Q_{\frac {a-4}{2}}^{\frac {a-2}{2}}(i x)\right ) \]