Internal problem ID [9013]
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]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\frac {b \cos \relax (x ) y^{\prime }}{\sin \relax (x ) a}+\frac {\left (c \left (\cos ^{2}\relax (x )\right )+d \cos \relax (x )+e \right ) y}{a \sin \relax (x )^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.232 (sec). Leaf size: 567
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)
\[ y \relax (x ) = c_{1} \left (\sin ^{-\frac {a +b}{2 a}}\relax (x )\right ) \hypergeom \left (\left [\frac {\sqrt {a^{2}+\left (-2 b -4 c -4 d -4 e \right ) a +b^{2}}+2 i \sqrt {4 c a -b^{2}}-\sqrt {a^{2}+\left (-2 b -4 c +4 d -4 e \right ) a +b^{2}}+2 a}{4 a}, -\frac {2 i \sqrt {4 c a -b^{2}}+\sqrt {a^{2}+\left (-2 b -4 c +4 d -4 e \right ) a +b^{2}}-\sqrt {a^{2}+\left (-2 b -4 c -4 d -4 e \right ) a +b^{2}}-2 a}{4 a}\right ], \left [-\frac {-2 a +\sqrt {a^{2}+\left (-2 b -4 c +4 d -4 e \right ) a +b^{2}}}{2 a}\right ], \frac {\cos \relax (x )}{2}+\frac {1}{2}\right ) \left (\frac {\cos \relax (x )}{2}-\frac {1}{2}\right )^{\frac {2 a +\sqrt {a^{2}+\left (-2 b -4 c -4 d -4 e \right ) a +b^{2}}}{4 a}} \left (2 \cos \relax (x )+2\right )^{-\frac {-2 a +\sqrt {a^{2}+\left (-2 b -4 c +4 d -4 e \right ) a +b^{2}}}{4 a}}+c_{2} \left (\sin ^{-\frac {a +b}{2 a}}\relax (x )\right ) \hypergeom \left (\left [\frac {\sqrt {a^{2}+\left (-2 b -4 c -4 d -4 e \right ) a +b^{2}}-2 i \sqrt {4 c a -b^{2}}+\sqrt {a^{2}+\left (-2 b -4 c +4 d -4 e \right ) a +b^{2}}+2 a}{4 a}, \frac {\sqrt {a^{2}+\left (-2 b -4 c -4 d -4 e \right ) a +b^{2}}+2 i \sqrt {4 c a -b^{2}}+\sqrt {a^{2}+\left (-2 b -4 c +4 d -4 e \right ) a +b^{2}}+2 a}{4 a}\right ], \left [\frac {2 a +\sqrt {a^{2}+\left (-2 b -4 c +4 d -4 e \right ) a +b^{2}}}{2 a}\right ], \frac {\cos \relax (x )}{2}+\frac {1}{2}\right ) \left (\frac {\cos \relax (x )}{2}-\frac {1}{2}\right )^{\frac {2 a +\sqrt {a^{2}+\left (-2 b -4 c -4 d -4 e \right ) a +b^{2}}}{4 a}} \left (2 \cos \relax (x )+2\right )^{\frac {2 a +\sqrt {a^{2}+\left (-2 b -4 c +4 d -4 e \right ) a +b^{2}}}{4 a}} \]
✓ Solution by Mathematica
Time used: 42.051 (sec). Leaf size: 1596424
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]
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