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 = 19.1912 (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 = 0.635 (sec), leaf count = 121

\[ \left \{ y \left ( x \right ) ={{\rm e}^{-x}} \left ( 1+x \right ) \left ( {\it \_C2}\,\int \!{\frac {x}{ \left ( 1+x \right ) ^{2}}{\it HeunC} \left ( i\sqrt {3},1,-2,-{\frac {3}{2}},{\frac {7}{2}},-x \right ) {{\rm e}^{-{\frac {x \left ( i\sqrt {3}-3 \right ) }{2}}}}}\,{\rm d}x+{\it \_C3}\,\int \!{\frac {x}{ \left ( 1+x \right ) ^{2}}{\it HeunC} \left ( i\sqrt {3},1,-2,-{\frac {3}{2}},{\frac {7}{2}},-x \right ) {{\rm e}^{-{\frac {x \left ( i\sqrt {3}-3 \right ) }{2}}}}\int \!{\frac { \left ( 1+x \right ) {{\rm e}^{i\sqrt {3}x}}}{{x}^{2}} \left ( {\it HeunC} \left ( i\sqrt {3},1,-2,-{\frac {3}{2}},{\frac {7}{2}},-x \right ) \right ) ^{-2}}\,{\rm d}x}\,{\rm d}x+{\it \_C1} \right ) \right \} \] 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),'implicit')

Maple raw output

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