Internal problem ID [5138]
Book: THEORY OF DIFFERENTIAL EQUATIONS IN ENGINEERING AND MECHANICS. K.T.
CHAU, CRC Press. Boca Raton, FL. 2018
Section: Chapter 3. Ordinary Differential Equations. Section 3.6 Summary and Problems. Page
218
Problem number: Problem 3.24.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (-x^{2}+1\right ) \eta ^{\prime \prime }-\left (1+x \right ) \eta ^{\prime }+\left (k +1\right ) \eta =0} \end {gather*}
✓ Solution by Maple
Time used: 0.65 (sec). Leaf size: 95
dsolve((1-x^2)*diff(eta(x),x$2)-(1+x)*diff(eta(x),x)+(k+1)*eta(x)=0,eta(x), singsol=all)
\[ \eta \relax (x ) = c_{1} \left (x +1\right )^{\sqrt {k +1}} \hypergeom \left (\left [-\sqrt {k +1}, 1-\sqrt {k +1}\right ], \left [1-2 \sqrt {k +1}\right ], \frac {2}{x +1}\right )+c_{2} \left (x +1\right )^{-\sqrt {k +1}} \hypergeom \left (\left [\sqrt {k +1}, 1+\sqrt {k +1}\right ], \left [1+2 \sqrt {k +1}\right ], \frac {2}{x +1}\right ) \]
✓ Solution by Mathematica
Time used: 0.029 (sec). Leaf size: 77
DSolve[(1-x^2)*z''[x]-(1+x)*z'[x]+(k+1)*z[x]==0,z[x],x,IncludeSingularSolutions -> True]
\begin{align*} z(x)\to c_2 G_{2,2}^{2,0}\left (\frac {1-x}{2}| {c} 1-\sqrt {k+1},\sqrt {k+1}+1 \\ 0,0 \\ \\ \right )+c_1 \, _2F_1\left (-\sqrt {k+1},\sqrt {k+1};1;\frac {1-x}{2}\right ) \\ \end{align*}