60.3.362 problem 1368

Internal problem ID [11372]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1368
Date solved : Tuesday, January 28, 2025 at 06:04:38 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.121 (sec). Leaf size: 69

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

\[ y = \left (x^{2}+1\right )^{\frac {1}{2}-\frac {a}{4}} \left (\operatorname {LegendreP}\left (\frac {a}{2}-1, \frac {\sqrt {a^{2}-4 a +4 b +4}}{2}, i x \right ) c_{1} +\operatorname {LegendreQ}\left (\frac {a}{2}-1, \frac {\sqrt {a^{2}-4 a +4 b +4}}{2}, i x \right ) c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.063 (sec). Leaf size: 92

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

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