48.4.16 problem Problem 3.23

Internal problem ID [7558]
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
Date solved : Tuesday, January 28, 2025 at 03:16:31 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (-x^{2}+1\right ) z^{\prime \prime }+\left (1-3 x \right ) z^{\prime }+k z&=0 \end{align*}

✓ Solution by Maple

Time used: 0.056 (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)
 

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

✓ Solution by Mathematica

Time used: 0.279 (sec). Leaf size: 77

DSolve[(1-x^2)*D[z[x],{x,2}]+(1-3*x)*D[z[x],x]+k*z[x]==0,z[x],x,IncludeSingularSolutions -> True]
 

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