ODE
\[ y(x) \left (a+b \sin ^2(x)\right )+y''(x)=0 \] ODE Classification
[_ellipsoidal]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0327791 (sec), leaf count = 40
\[\left \{\left \{y(x)\to c_1 \text {MathieuC}\left [a+\frac {b}{2},\frac {b}{4},x\right ]+c_2 \text {MathieuS}\left [a+\frac {b}{2},\frac {b}{4},x\right ]\right \}\right \}\]
Maple ✓
cpu = 0.302 (sec), leaf count = 29
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{\it MathieuC} \left ( {\frac {b}{2}}+a,{\frac {b}{4}},x \right ) +{\it \_C2}\,{\it MathieuS} \left ( {\frac {b}{2}}+a,{\frac {b}{4}},x \right ) \right \} \] Mathematica raw input
DSolve[(a + b*Sin[x]^2)*y[x] + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1]*MathieuC[a + b/2, b/4, x] + C[2]*MathieuS[a + b/2, b/4, x]}}
Maple raw input
dsolve(diff(diff(y(x),x),x)+(a+b*sin(x)^2)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = _C1*MathieuC(1/2*b+a,1/4*b,x)+_C2*MathieuS(1/2*b+a,1/4*b,x)