ODE
\[ y(x) \left (\text {a0}+4 \text {a1} \cosh ^2(x)-\text {a2} \text {sech}^2(x)\right )+y''(x)+\tanh (x) y'(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✗
cpu = 2.1481 (sec), leaf count = 0 , could not solve
DSolve[(a0 + 4*a1*Cosh[x]^2 - a2*Sech[x]^2)*y[x] + Tanh[x]*Derivative[1][y][x] + Derivative[2][y][x] == 0, y[x], x]
Maple ✓
cpu = 1.084 (sec), leaf count = 71
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{\it HeunC} \left ( 0,i\sqrt {{\it a2}},-{\frac {1}{2}},{\it a1},{\frac {1}{4}}+{\frac {{\it a0}}{4}}-{\frac {{\it a2}}{4}}, \left ( \cosh \left ( x \right ) \right ) ^{2} \right ) \left ( \cosh \left ( x \right ) \right ) ^{i\sqrt {{\it a2}}}+{\it \_C2}\,{\it HeunC} \left ( 0,-i\sqrt {{\it a2}},-{\frac {1}{2}},{\it a1},{\frac {1}{4}}+{\frac {{\it a0}}{4}}-{\frac {{\it a2}}{4}}, \left ( \cosh \left ( x \right ) \right ) ^{2} \right ) \left ( \cosh \left ( x \right ) \right ) ^{-i\sqrt {{\it a2}}} \right \} \] Mathematica raw input
DSolve[(a0 + 4*a1*Cosh[x]^2 - a2*Sech[x]^2)*y[x] + Tanh[x]*y'[x] + y''[x] == 0,y[x],x]
Mathematica raw output
DSolve[(a0 + 4*a1*Cosh[x]^2 - a2*Sech[x]^2)*y[x] + Tanh[x]*Derivative[1][y][x] + Derivative[2][y][x] == 0, y[x], x]
Maple raw input
dsolve(diff(diff(y(x),x),x)+diff(y(x),x)*tanh(x)+(a0+4*a1*cosh(x)^2-a2*sech(x)^2)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = _C1*HeunC(0,I*a2^(1/2),-1/2,a1,1/4+1/4*a0-1/4*a2,cosh(x)^2)*cosh(x)^(I*a2^(1/2))+_C2*HeunC(0,-I*a2^(1/2),-1/2,a1,1/4+1/4*a0-1/4*a2,cosh(x)^2)*cosh(x)^(-I*a2^(1/2))