ODE
\[ y(x) \left (\text {a0}+\text {a1} x^2+x^4\right )+y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.210719 (sec), leaf count = 100
\[\left \{\left \{y(x)\to c_1{}^2 e^{\frac {1}{6} i x \left (3 \text {a1}+2 x^2\right )} \text {HeunT}\left [\frac {\text {a1}^2}{4}-\text {a0},2 i,i \text {a1},0,2 i,x\right ]+c_2 e^{-\frac {1}{6} i x \left (3 \text {a1}+2 x^2\right )} \text {HeunT}\left [\frac {\text {a1}^2}{4}-\text {a0},-2 i,-i \text {a1},0,-2 i,x\right ]\right \}\right \}\]
Maple ✓
cpu = 1.591 (sec), leaf count = 109
\[\left [y \left (x \right ) = \textit {\_C1} \,{\mathrm e}^{\frac {i x \left (2 x^{2}+3 \mathit {a1} \right )}{6}} \mathit {HT}\left (\frac {\left (\mathit {a1}^{2}-4 \mathit {a0} \right ) 3^{\frac {2}{3}} 2^{\frac {1}{3}}}{8}, 0, -\frac {\mathit {a1} 2^{\frac {2}{3}} 3^{\frac {1}{3}}}{2}, \frac {i 2^{\frac {1}{3}} 3^{\frac {2}{3}} x}{3}\right )+\textit {\_C2} \mathit {HT}\left (\frac {\left (\mathit {a1}^{2}-4 \mathit {a0} \right ) 3^{\frac {2}{3}} 2^{\frac {1}{3}}}{8}, 0, -\frac {\mathit {a1} 2^{\frac {2}{3}} 3^{\frac {1}{3}}}{2}, -\frac {i 2^{\frac {1}{3}} 3^{\frac {2}{3}} x}{3}\right ) {\mathrm e}^{-\frac {i x \left (2 x^{2}+3 \mathit {a1} \right )}{6}}\right ]\] Mathematica raw input
DSolve[(a0 + a1*x^2 + x^4)*y[x] + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (C[2]*HeunT[-a0 + a1^2/4, -2*I, (-I)*a1, 0, -2*I, x])/E^((I/6)*x*(3*a1 + 2*x^2)) + E^((I/6)*x*(3*a1 + 2*x^2))*C[1]^2*HeunT[-a0 + a1^2/4, 2*I, I*a1, 0, 2*I, x]}}
Maple raw input
dsolve(diff(diff(y(x),x),x)+(x^4+a1*x^2+a0)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*exp(1/6*I*x*(2*x^2+3*a1))*HeunT(1/8*(a1^2-4*a0)*3^(2/3)*2^(1/3),0,-1/2*a1*2^(2/3)*3^(1/3),1/3*I*2^(1/3)*3^(2/3)*x)+_C2*HeunT(1/8*(a1^2-4*a0)*3^(2/3)*2^(1/3),0,-1/2*a1*2^(2/3)*3^(1/3),-1/3*I*2^(1/3)*3^(2/3)*x)*exp(-1/6*I*x*(2*x^2+3*a1))]