Internal problem ID [8960]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1381.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\frac {b y}{x^{2} \left (x -a \right )^{2}}-c=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 219
dsolve(diff(diff(y(x),x),x) = -b/x^2/(x-a)^2*y(x)+c,y(x), singsol=all)
\[ y \relax (x ) = \sqrt {x \left (-x +a \right )}\, \left (\frac {-x +a}{x}\right )^{\frac {\sqrt {a^{2}-4 b}}{2 a}} c_{2}+\sqrt {x \left (-x +a \right )}\, \left (\frac {x}{-x +a}\right )^{\frac {\sqrt {a^{2}-4 b}}{2 a}} c_{1}+\frac {\left (-\left (\int \sqrt {x \left (-x +a \right )}\, \left (\frac {-x +a}{x}\right )^{-\frac {\sqrt {a^{2}-4 b}}{2 a}}d x \right ) \left (\frac {-x +a}{x}\right )^{\frac {\sqrt {a^{2}-4 b}}{2 a}}+\left (\int \sqrt {x \left (-x +a \right )}\, \left (\frac {x}{-x +a}\right )^{-\frac {\sqrt {a^{2}-4 b}}{2 a}}d x \right ) \left (\frac {x}{-x +a}\right )^{\frac {\sqrt {a^{2}-4 b}}{2 a}}\right ) \sqrt {x \left (-x +a \right )}\, c}{\sqrt {a^{2}-4 b}} \]
✓ Solution by Mathematica
Time used: 0.39 (sec). Leaf size: 273
DSolve[y''[x] == c - (b*y[x])/(x^2*(-a + x)^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c x^2 \left (a^2 \left (3 \sqrt {1-\frac {4 b}{a^2}}+1\right )-4 b\right ) (a-x)^2 \, _2F_1\left (1,3;\frac {5}{2}-\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}};\frac {x}{a}\right )-c x^2 \left (a^2 \left (3 \sqrt {1-\frac {4 b}{a^2}}-1\right )+4 b\right ) (a-x)^2 \, _2F_1\left (1,3;\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}}+\frac {5}{2};\frac {x}{a}\right )+2 \left (2 a^2+b\right ) x^{\frac {1}{2}-\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}}} (x-a)^{\frac {1}{2}-\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}}} \left (c_1 \left (a^2-4 b\right ) x^{\sqrt {1-\frac {4 b}{a^2}}}+a c_2 \sqrt {1-\frac {4 b}{a^2}} (x-a)^{\sqrt {1-\frac {4 b}{a^2}}}\right )}{2 \left (a^2-4 b\right ) \left (2 a^2+b\right )} \\ \end{align*}