2.1070   ODE No. 1070

\[ a \tan (x) y'(x)+b y(x)+y''(x)=0 \]

✓ Mathematica : cpu = 0.166771 (sec), leaf count = 143

DSolve[b*y[x] + a*Tan[x]*Derivative[1][y][x] + Derivative[2][y][x] == 0,y[x],x]
 

\[\left \{\left \{y(x)\to c_1 \, _2F_1\left (-\frac {a}{4}-\frac {1}{4} \sqrt {a^2+4 b},\frac {1}{4} \sqrt {a^2+4 b}-\frac {a}{4};\frac {1}{2}-\frac {a}{2};\cos ^2(x)\right )+i^{a+1} c_2 \cos ^{a+1}(x) \, _2F_1\left (\frac {a}{4}-\frac {1}{4} \sqrt {a^2+4 b}+\frac {1}{2},\frac {a}{4}+\frac {1}{4} \sqrt {a^2+4 b}+\frac {1}{2};\frac {a}{2}+\frac {3}{2};\cos ^2(x)\right )\right \}\right \}\]

✓ Maple : cpu = 0.267 (sec), leaf count = 60

dsolve(diff(diff(y(x),x),x)+a*diff(y(x),x)*tan(x)+b*y(x)=0,y(x))
 

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