48.4.17 problem Problem 3.24

Internal problem ID [7559]
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
Date solved : Tuesday, January 28, 2025 at 03:16:32 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (-x^{2}+1\right ) \eta ^{\prime \prime }-\left (1+x \right ) \eta ^{\prime }+\left (k +1\right ) \eta &=0 \end{align*}

✓ Solution by Maple

Time used: 0.056 (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)
 

\[ \eta = c_{1} \left (x +1\right )^{\sqrt {k +1}} \operatorname {hypergeom}\left (\left [-\sqrt {k +1}, 1-\sqrt {k +1}\right ], \left [1-2 \sqrt {k +1}\right ], \frac {2}{x +1}\right )+c_{2} \left (x +1\right )^{-\sqrt {k +1}} \operatorname {hypergeom}\left (\left [\sqrt {k +1}, 1+\sqrt {k +1}\right ], \left [1+2 \sqrt {k +1}\right ], \frac {2}{x +1}\right ) \]

✓ Solution by Mathematica

Time used: 0.395 (sec). Leaf size: 77

DSolve[(1-x^2)*D[z[x],{x,2}]-(1+x)*D[z[x],x]+(k+1)*z[x]==0,z[x],x,IncludeSingularSolutions -> True]
 

\[ z(x)\to c_2 G_{2,2}^{2,0}\left (\frac {1-x}{2}| \begin {array}{c} 1-\sqrt {k+1},\sqrt {k+1}+1 \\ 0,0 \\ \end {array} \right )+c_1 \operatorname {Hypergeometric2F1}\left (-\sqrt {k+1},\sqrt {k+1},1,\frac {1-x}{2}\right ) \]