4.44.7 \(x y'''(x)+3 y'(x)+x y(x)=0\)

ODE
\[ x y'''(x)+3 y'(x)+x y(x)=0 \] ODE Classification

[[_3rd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✗
cpu = 20.3335 (sec), leaf count = 0 , DifferentialRoot result

\[\left \{\left \{y(x)\to \text {DifferentialRoot}\left [\{\unicode {f818},\unicode {f817}\}\unicode {f4a1}\left \{\unicode {f817} \unicode {f818}^{\text {Symbol}[\text {StringJoin}[\text {ConstantArray}[\prime ,3]]]}(\unicode {f817})+3 \unicode {f818}'(\unicode {f817})+\unicode {f817} \unicode {f818}(\unicode {f817})=0,\unicode {f818}(1)=c_1,\unicode {f818}'(1)=c_2,\unicode {f818}''(1)=c_3\right \}\right ][x]\right \}\right \}\]

Maple ✓
cpu = 2.29 (sec), leaf count = 135

\[\left [y \left (x \right ) = \textit {\_C1} \left (x +1\right ) {\mathrm e}^{-x}+\textit {\_C2} \left (\int \frac {\HeunC \left (i \sqrt {3}, 1, -2, -\frac {3}{2}, \frac {7}{2}, -x \right ) x \,{\mathrm e}^{-\frac {x \left (i \sqrt {3}-3\right )}{2}}}{\left (x +1\right )^{2}}d x \right ) {\mathrm e}^{-x} \left (x +1\right )+\textit {\_C3} \left (\int \frac {x \HeunC \left (i \sqrt {3}, 1, -2, -\frac {3}{2}, \frac {7}{2}, -x \right ) {\mathrm e}^{-\frac {x \left (i \sqrt {3}-3\right )}{2}} \left (\int \frac {\left (x +1\right ) {\mathrm e}^{i \sqrt {3}\, x}}{\HeunC \left (i \sqrt {3}, 1, -2, -\frac {3}{2}, \frac {7}{2}, -x \right )^{2} x^{2}}d x \right )}{\left (x +1\right )^{2}}d x \right ) {\mathrm e}^{-x} \left (x +1\right )\right ]\] Mathematica raw input

DSolve[x*y[x] + 3*y'[x] + x*y'''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> DifferentialRoot[Function[{\[FormalY], \[FormalX]}, {\[FormalX]*\[Form
alY][\[FormalX]] + 3*Derivative[1][\[FormalY]][\[FormalX]] + \[FormalX]*Derivati
ve[3][\[FormalY]][\[FormalX]] == 0, \[FormalY][1] == C[1], Derivative[1][\[Forma
lY]][1] == C[2], Derivative[2][\[FormalY]][1] == C[3]}]][x]}}

Maple raw input

dsolve(x*diff(diff(diff(y(x),x),x),x)+3*diff(y(x),x)+x*y(x) = 0, y(x))

Maple raw output

[y(x) = _C1*(x+1)*exp(-x)+_C2*Int(HeunC(I*3^(1/2),1,-2,-3/2,7/2,-x)*x*exp(-1/2*x
*(I*3^(1/2)-3))/(x+1)^2,x)*exp(-x)*(x+1)+_C3*Int(x*HeunC(I*3^(1/2),1,-2,-3/2,7/2
,-x)*exp(-1/2*x*(I*3^(1/2)-3))*Int(1/HeunC(I*3^(1/2),1,-2,-3/2,7/2,-x)^2*(x+1)/x
^2*exp(I*3^(1/2)*x),x)/(x+1)^2,x)*exp(-x)*(x+1)]