54.3.348 problem 1365

Internal problem ID [12643]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1365
Date solved : Wednesday, October 01, 2025 at 02:18:52 AM
CAS classification : [_Halm]

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 55

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

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

✓ Mathematica. Time used: 0.057 (sec). Leaf size: 86

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

\begin{align*} y(x)&\to \exp \left (\int _1^x\frac {K[1]+i \sqrt {a+1}}{K[1]^2+1}dK[1]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[2]}\frac {K[1]+i \sqrt {a+1}}{K[1]^2+1}dK[1]\right )dK[2]+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.234 (sec). Leaf size: 97

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a*y(x)/(x**2 + 1)**2 + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {\sqrt {x^{2} + 1} \left (C_{1} \sqrt {\frac {x^{2}}{x^{2} + 1}} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2} - \frac {\sqrt {a + 1}}{2}, \frac {\sqrt {a + 1}}{2} + \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {x^{2}}{x^{2} + 1}} \right )} + C_{2} {{}_{2}F_{1}\left (\begin {matrix} - \frac {\sqrt {a + 1}}{2}, \frac {\sqrt {a + 1}}{2} \\ \frac {1}{2} \end {matrix}\middle | {\frac {x^{2}}{x^{2} + 1}} \right )}\right ) \sqrt [4]{x^{2}}}{\sqrt {x}} \]