60.3.224 problem 1228

Internal problem ID [11234]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1228
Date solved : Monday, January 27, 2025 at 10:49:32 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}+1\right ) y^{\prime \prime }+3 x y^{\prime }+a y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.138 (sec). Leaf size: 53

dsolve((x^2+1)*diff(diff(y(x),x),x)+3*x*diff(y(x),x)+a*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.043 (sec). Leaf size: 66

DSolve[a*y[x] + 3*x*D[y[x],x] + (1 + x^2)*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

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