60.3.376 problem 1382

Internal problem ID [11386]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1382
Date solved : Monday, January 27, 2025 at 11:18:51 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.011 (sec). Leaf size: 104

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

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

✓ Solution by Mathematica

Time used: 0.593 (sec). Leaf size: 168

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

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