Internal problem ID [8962]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1383.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\frac {\left (\left (\alpha +\beta +1\right ) \left (x -a \right )^{2} \left (x -b \right )+\left (1-\alpha -\beta \right ) \left (x -b \right )^{2} \left (x -a \right )\right ) y^{\prime }}{\left (x -a \right )^{2} \left (x -b \right )^{2}}+\frac {\alpha \beta \left (-b +a \right )^{2} y}{\left (x -a \right )^{2} \left (x -b \right )^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 39
dsolve(diff(diff(y(x),x),x) = -((alpha+beta+1)*(x-a)^2*(x-b)+(1-alpha-beta)*(x-b)^2*(x-a))/(x-a)^2/(x-b)^2*diff(y(x),x)-alpha*beta*(a-b)^2/(x-a)^2/(x-b)^2*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \left (\frac {-x +a}{b -x}\right )^{\beta }+c_{2} \left (\frac {-x +a}{b -x}\right )^{\alpha } \]
✓ Solution by Mathematica
Time used: 0.063 (sec). Leaf size: 44
DSolve[y''[x] == -((\[Alpha]*(a - b)^2*\[Beta]*y[x])/((-a + x)^2*(-b + x)^2)) - (((1 + \[Alpha] + \[Beta])*(-a + x)^2*(-b + x) + (1 - \[Alpha] - \[Beta])*(-a + x)*(-b + x)^2)*y'[x])/((-a + x)^2*(-b + x)^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 (x-a)^{\alpha } (x-b)^{-\alpha }+c_2 (x-a)^{\beta } (x-b)^{-\beta } \\ \end{align*}