60.3.379 problem 1385

Internal problem ID [11389]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1385
Date solved : Monday, January 27, 2025 at 11:19:01 PM
CAS classification : [_Halm]

\begin{align*} y^{\prime \prime }&=-\frac {\left (a \,x^{2}+a -3\right ) y}{4 \left (x^{2}+1\right )^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 55

dsolve(diff(diff(y(x),x),x) = -1/4*(a*x^2+a-3)/(x^2+1)^2*y(x),y(x), singsol=all)
 

\[ y = \left (x^{2}+1\right )^{{1}/{4}} \left (\left (x +\sqrt {x^{2}+1}\right )^{-\frac {\sqrt {-a +1}}{2}} c_{2} +\left (x +\sqrt {x^{2}+1}\right )^{\frac {\sqrt {-a +1}}{2}} c_{1} \right ) \]

✓ Solution by Mathematica

Time used: 0.054 (sec). Leaf size: 70

DSolve[D[y[x],{x,2}] == -1/4*((-3 + a + a*x^2)*y[x])/(1 + x^2)^2,y[x],x,IncludeSingularSolutions -> True]
 

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