60.3.414 problem 1420

Internal problem ID [11424]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1420
Date solved : Tuesday, January 28, 2025 at 06:05:26 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \cos \left (x \right )^{2} y^{\prime \prime }-\left (a \cos \left (x \right )^{2}+n \left (n -1\right )\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 6.926 (sec). Leaf size: 110

dsolve(cos(x)^2*diff(diff(y(x),x),x)-(a*cos(x)^2+n*(n-1))*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {\sin \left (x \right )^{{3}/{2}} \left (\left (\frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )^{\frac {3}{4}-\frac {n}{2}} \operatorname {hypergeom}\left (\left [1+\frac {i \sqrt {a}}{2}-\frac {n}{2}, 1-\frac {i \sqrt {a}}{2}-\frac {n}{2}\right ], \left [\frac {3}{2}-n \right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) c_{2} +\cos \left (x \right )^{n +\frac {1}{2}} \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {i \sqrt {a}}{2}+\frac {n}{2}, \frac {1}{2}-\frac {i \sqrt {a}}{2}+\frac {n}{2}\right ], \left [n +\frac {1}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) c_{1} \right )}{\sqrt {\sin \left (2 x \right )}} \]

✓ Solution by Mathematica

Time used: 0.570 (sec). Leaf size: 126

DSolve[(-((-1 + n)*n) - a*Cos[x]^2)*y[x] + Cos[x]^2*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_1 i^{1-n} \cos ^{1-n}(x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (-n-i \sqrt {a}+1\right ),\frac {1}{2} \left (-n+i \sqrt {a}+1\right ),\frac {3}{2}-n,\cos ^2(x)\right )+c_2 i^n \cos ^n(x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (n-i \sqrt {a}\right ),\frac {1}{2} \left (n+i \sqrt {a}\right ),n+\frac {1}{2},\cos ^2(x)\right ) \]