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 = 0.723731 (sec), leaf count = 0 , DifferentialRoot result
\[\left \{\left \{y(x)\to \text {DifferentialRoot}\left (\{\unicode {f818},\unicode {f817}\}\unicode {f4a1}\left \{\left (\unicode {f817}^4+\text {a1} \unicode {f817}^2+\text {a0}\right ) \unicode {f818}(\unicode {f817})+\unicode {f818}''(\unicode {f817})=0,\unicode {f818}(0)=c_1,\unicode {f818}'(0)=c_2\right \}\right )(x)\right \}\right \}\]
Maple ✓
cpu = 0.294 (sec), leaf count = 109
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{{\rm e}^{{\frac {i}{6}}x \left ( 2\,{x}^{2}+3\,{\it a1} \right ) }}{\it HeunT} \left ( {\frac { \left ( {{\it a1}}^{2}-4\,{\it a0} \right ) {3}^{{\frac {2}{3}}}\sqrt [3]{2}}{8}},0,-{\frac {{\it a1}\,{2}^{{\frac {2}{3}}}\sqrt [3]{3}}{2}},{\frac {i}{3}}\sqrt [3]{2}{3}^{{\frac {2}{3}}}x \right ) +{\it \_C2}\,{\it HeunT} \left ( {\frac { \left ( {{\it a1}}^{2}-4\,{\it a0} \right ) {3}^{{\frac {2}{3}}}\sqrt [3]{2}}{8}},0,-{\frac {{\it a1}\,{2}^{{\frac {2}{3}}}\sqrt [3]{3}}{2}},-{\frac {i}{3}}\sqrt [3]{2}{3}^{{\frac {2}{3}}}x \right ) {{\rm e}^{-{\frac {i}{6}}x \left ( 2\,{x}^{2}+3\,{\it a1} \right ) }} \right \} \] Mathematica raw input
DSolve[(a0 + a1*x^2 + x^4)*y[x] + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> DifferentialRoot[Function[{\[FormalY], \[FormalX]}, {(\[FormalX]^4 + a0 + \[FormalX]^2*a1)*\[FormalY][\[FormalX]] + Derivative[2][\[FormalY]][\[FormalX]] == 0, \[FormalY][0] == C[1], Derivative[1][\[FormalY]][0] == C[2]}]][x]}}
Maple raw input
dsolve(diff(diff(y(x),x),x)+(x^4+a1*x^2+a0)*y(x) = 0, y(x),'implicit')
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))