60.3.351 problem 1357

Internal problem ID [11361]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1357
Date solved : Tuesday, January 28, 2025 at 06:04:25 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.291 (sec). Leaf size: 97

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

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

✓ Solution by Mathematica

Time used: 0.709 (sec). Leaf size: 264

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

\[ y(x)\to x^{-\frac {1}{2} \sqrt {a^2-4 a-4 c+4}-\frac {a}{2}+1} \left (c_1 \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (-\sqrt {a^2-2 a-4 b+1}-\sqrt {a^2-4 a-4 c+4}+1\right ),\frac {1}{4} \left (\sqrt {a^2-2 a-4 b+1}-\sqrt {a^2-4 a-4 c+4}+1\right ),1-\frac {1}{2} \sqrt {a^2-4 a-4 c+4},-x^2\right )+c_2 x^{\sqrt {a^2-4 a-4 c+4}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (-\sqrt {a^2-2 a-4 b+1}+\sqrt {a^2-4 a-4 c+4}+1\right ),\frac {1}{4} \left (\sqrt {a^2-2 a-4 b+1}+\sqrt {a^2-4 a-4 c+4}+1\right ),\frac {1}{2} \left (\sqrt {a^2-4 a-4 c+4}+2\right ),-x^2\right )\right ) \]