ODE
\[ a \tan (x) y'(x)+b y(x)+y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.306397 (sec), leaf count = 129
\[\left \{\left \{y(x)\to c_1 \, _2F_1\left (\frac {1}{4} \left (-a-\sqrt {a^2+4 b}\right ),\frac {1}{4} \left (\sqrt {a^2+4 b}-a\right );\frac {1-a}{2};\cos ^2(x)\right )+i^{a+1} c_2 \cos ^{a+1}(x) \, _2F_1\left (\frac {1}{4} \left (a-\sqrt {a^2+4 b}+2\right ),\frac {1}{4} \left (a+\sqrt {a^2+4 b}+2\right );\frac {a+3}{2};\cos ^2(x)\right )\right \}\right \}\]
Maple ✓
cpu = 0.185 (sec), leaf count = 60
\[ \left \{ y \left ( x \right ) = \left ( \cos \left ( x \right ) \right ) ^{{\frac {1}{2}}+{\frac {a}{2}}} \left ( {\it LegendreQ} \left ( {\frac {1}{2}\sqrt {{a}^{2}+4\,b}}-{\frac {1}{2}},{\frac {1}{2}}+{\frac {a}{2}},\sin \left ( x \right ) \right ) {\it \_C2}+{\it LegendreP} \left ( {\frac {1}{2}\sqrt {{a}^{2}+4\,b}}-{\frac {1}{2}},{\frac {1}{2}}+{\frac {a}{2}},\sin \left ( x \right ) \right ) {\it \_C1} \right ) \right \} \] Mathematica raw input
DSolve[b*y[x] + a*Tan[x]*y'[x] + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1]*Hypergeometric2F1[(-a - Sqrt[a^2 + 4*b])/4, (-a + Sqrt[a^2 + 4*b])/4, (1 - a)/2, Cos[x]^2] + I^(1 + a)*C[2]*Cos[x]^(1 + a)*Hypergeometric2F1[(2 + a - Sqrt[a^2 + 4*b])/4, (2 + a + Sqrt[a^2 + 4*b])/4, (3 + a)/2, Cos[x]^2]}}
Maple raw input
dsolve(diff(diff(y(x),x),x)+a*diff(y(x),x)*tan(x)+b*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = cos(x)^(1/2+1/2*a)*(LegendreQ(1/2*(a^2+4*b)^(1/2)-1/2,1/2+1/2*a,sin(x))*_C2+LegendreP(1/2*(a^2+4*b)^(1/2)-1/2,1/2+1/2*a,sin(x))*_C1)