Internal problem ID [5137]
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 3.23.
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 ) z^{\prime \prime }+\left (1-3 x \right ) z^{\prime }+k z=0} \end {gather*}
✓ Solution by Maple
Time used: 0.152 (sec). Leaf size: 99
dsolve((1-x^2)*diff(z(x),x$2)+(1-3*x)*diff(z(x),x)+k*z(x)=0,z(x), singsol=all)
\[ z \relax (x ) = c_{1} \left (x +1\right )^{-1-\sqrt {k +1}} \hypergeom \left (\left [\sqrt {k +1}, 1+\sqrt {k +1}\right ], \left [1+2 \sqrt {k +1}\right ], \frac {2}{x +1}\right )+c_{2} \left (x +1\right )^{-1+\sqrt {k +1}} \hypergeom \left (\left [-\sqrt {k +1}, 1-\sqrt {k +1}\right ], \left [1-2 \sqrt {k +1}\right ], \frac {2}{x +1}\right ) \]
✓ Solution by Mathematica
Time used: 0.034 (sec). Leaf size: 77
DSolve[(1-x^2)*z''[x]+(1-3*x)*z'[x]+k*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} -\sqrt {k+1},\sqrt {k+1} \\ 0,0 \\ \\ \right )+c_1 \, _2F_1\left (1-\sqrt {k+1},\sqrt {k+1}+1;1;\frac {1-x}{2}\right ) \\ \end{align*}