60.3.388 problem 1394

Internal problem ID [11398]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1394
Date solved : Monday, January 27, 2025 at 11:19:19 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 79

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

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

✓ Solution by Mathematica

Time used: 2.718 (sec). Leaf size: 117

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

\[ y(x)\to \frac {\exp \left (\int _1^x\frac {\sqrt {1-\frac {4 c}{b^2}} b+b+2 a K[1]}{2 K[1] (b+a K[1])}dK[1]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[2]}\frac {\sqrt {1-\frac {4 c}{b^2}} b+b+2 a K[1]}{2 K[1] (b+a K[1])}dK[1]\right )dK[2]+c_1\right )}{x} \]