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 = 9.24189 (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 = 3.463 (sec), leaf count = 71
\[\left [y \left (x \right ) = \textit {\_C1} \HeunC \left (0, i \sqrt {\mathit {a2}}, -\frac {1}{2}, \mathit {a1} , \frac {1}{4}+\frac {\mathit {a0}}{4}-\frac {\mathit {a2}}{4}, \cosh ^{2}\left (x \right )\right ) \left (\cosh ^{i \sqrt {\mathit {a2}}}\left (x \right )\right )+\textit {\_C2} \HeunC \left (0, -i \sqrt {\mathit {a2}}, -\frac {1}{2}, \mathit {a1} , \frac {1}{4}+\frac {\mathit {a0}}{4}-\frac {\mathit {a2}}{4}, \cosh ^{2}\left (x \right )\right ) \left (\cosh ^{-i \sqrt {\mathit {a2}}}\left (x \right )\right )\right ]\] 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))
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))]