Internal problem ID [8840]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1260.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_y]]
Solve \begin {gather*} \boxed {x \left (x -1\right ) y^{\prime \prime }+\left (\left (\mathit {a1} +\mathit {b1} +1\right ) x -\mathit {d1} \right ) y^{\prime }+\mathit {a1} \mathit {b1} \mathit {d1}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 76
dsolve(x*(x-1)*diff(diff(y(x),x),x)+((a1+b1+1)*x-d1)*diff(y(x),x)+a1*b1*d1=0,y(x), singsol=all)
\[ y \relax (x ) = \int \left (-\hypergeom \left (\left [\mathit {d1} , -\mathit {a1} -\mathit {b1} +\mathit {d1} \right ], \left [1+\mathit {d1} \right ], x\right ) \mathrm {signum}\left (x -1\right )^{\mathit {a1} +\mathit {b1} -\mathit {d1}} \left (-\mathrm {signum}\left (x -1\right )\right )^{-\mathit {a1} -\mathit {b1} +\mathit {d1}} \mathit {a1} \mathit {b1} +x^{-\mathit {d1}} c_{1}\right ) \left (x -1\right )^{-\mathit {a1} -\mathit {b1} -1+\mathit {d1}}d x +c_{2} \]
✓ Solution by Mathematica
Time used: 0.35 (sec). Leaf size: 44
DSolve[a1*b1*d1 + (-d1 + (1 + a1 + b1)*x)*y'[x] + (-1 + x)*x*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {a1} \text {b1} x \, _3F_2(1,1,\text {a1}+\text {b1}+1;2,\text {d1}+1;x)+c_1 B_x(1-\text {d1},-\text {a1}-\text {b1}+\text {d1})+c_2 \\ \end{align*}