60.3.419 problem 1425

Internal problem ID [11429]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1425
Date solved : Tuesday, January 28, 2025 at 06:05:32 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {\left (-a^{2} \cos \left (x \right )^{2}-\left (3-2 a \right ) \cos \left (x \right )-3+3 a \right ) y}{\sin \left (x \right )^{2}} \end{align*}

✓ Solution by Maple

Time used: 1.371 (sec). Leaf size: 87

dsolve(diff(diff(y(x),x),x) = -(-a^2*cos(x)^2-(3-2*a)*cos(x)-3+3*a)/sin(x)^2*y(x),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 1.003 (sec). Leaf size: 125

DSolve[D[y[x],{x,2}] == (3 - 3*a + (3 - 2*a)*Cos[x] + a^2*Cos[x]^2)*Csc[x]^2*y[x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \exp \left (\int -\frac {\csc ^2(x) \left (a (2 a-1) \cos ^2(x)+(3-4 a) \cos (x)-2 a+3\right )}{(2 a-1) \cos (x)-2} \, d\cos (x)\right ) \left (c_2 \int _1^{\cos (x)}\exp \left (\int _1^{K[2]}\frac {-4 a^2 K[1]^2+K[1]^2-4 K[1]+a (8 K[1]+4)-6}{((2 a-1) K[1]-2) \left (K[1]^2-1\right )}dK[1]\right )dK[2]+c_1\right ) \]