60.3.371 problem 1377

Internal problem ID [11381]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1377
Date solved : Monday, January 27, 2025 at 11:18:38 PM
CAS classification : [[_Emden, _Fowler]]

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

✓ Solution by Maple

Time used: 0.011 (sec). Leaf size: 75

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

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

✓ Solution by Mathematica

Time used: 0.197 (sec). Leaf size: 104

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

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