Internal problem ID [9762]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1434.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\frac {b \cos \left (x \right ) y^{\prime }}{\sin \left (x \right ) a}+\frac {\left (c \cos \left (x \right )^{2}+d \cos \left (x \right )+e \right ) y}{a \sin \left (x \right )^{2}}=0} \]
✓ Solution by Maple
Time used: 0.875 (sec). Leaf size: 515
dsolve(diff(diff(y(x),x),x) = -b/sin(x)*cos(x)/a*diff(y(x),x)-(c*cos(x)^2+d*cos(x)+e)/a/sin(x)^2*y(x),y(x), singsol=all)
\[ y \left (x \right ) = \frac {\sqrt {2}\, \sin \left (x \right )^{-\frac {a +b}{2 a}} \sqrt {\cos \left (x \right )-1}\, \left (\frac {\cos \left (x \right )}{2}-\frac {1}{2}\right )^{\frac {\sqrt {a^{2}+\left (-2 b -4 c -4 d -4 e \right ) a +b^{2}}}{4 a}} \left (c_{1} \cos \left (\frac {x}{2}\right )^{-\frac {-2 a +\sqrt {a^{2}+\left (-2 b -4 c +4 d -4 e \right ) a +b^{2}}}{2 a}} \operatorname {hypergeom}\left (\left [\frac {\sqrt {a^{2}+\left (-2 b -4 c -4 d -4 e \right ) a +b^{2}}-2 i \sqrt {4 a c -b^{2}}-\sqrt {a^{2}+\left (-2 b -4 c +4 d -4 e \right ) a +b^{2}}+2 a}{4 a}, \frac {\sqrt {a^{2}+\left (-2 b -4 c -4 d -4 e \right ) a +b^{2}}+2 i \sqrt {4 a c -b^{2}}-\sqrt {a^{2}+\left (-2 b -4 c +4 d -4 e \right ) a +b^{2}}+2 a}{4 a}\right ], \left [1-\frac {\sqrt {a^{2}+\left (-2 b -4 c +4 d -4 e \right ) a +b^{2}}}{2 a}\right ], \frac {\cos \left (x \right )}{2}+\frac {1}{2}\right )+c_{2} \cos \left (\frac {x}{2}\right )^{\frac {2 a +\sqrt {a^{2}+\left (-2 b -4 c +4 d -4 e \right ) a +b^{2}}}{2 a}} \operatorname {hypergeom}\left (\left [\frac {\sqrt {a^{2}+\left (-2 b -4 c -4 d -4 e \right ) a +b^{2}}-2 i \sqrt {4 a c -b^{2}}+\sqrt {a^{2}+\left (-2 b -4 c +4 d -4 e \right ) a +b^{2}}+2 a}{4 a}, \frac {\sqrt {a^{2}+\left (-2 b -4 c -4 d -4 e \right ) a +b^{2}}+2 i \sqrt {4 a c -b^{2}}+\sqrt {a^{2}+\left (-2 b -4 c +4 d -4 e \right ) a +b^{2}}+2 a}{4 a}\right ], \left [1+\frac {\sqrt {a^{2}+\left (-2 b -4 c +4 d -4 e \right ) a +b^{2}}}{2 a}\right ], \frac {\cos \left (x \right )}{2}+\frac {1}{2}\right )\right )}{2} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y''[x] == -(((e + d*Cos[x] + c*Cos[x]^2)*Csc[x]^2*y[x])/a) - (b*Cot[x]*y'[x])/a,y[x],x,IncludeSingularSolutions -> True]
Timed out