4.35.16 \(x^4 y''(x)+\left (x^2+1\right ) x y'(x)+y(x)=0\)

ODE
\[ x^4 y''(x)+\left (x^2+1\right ) x y'(x)+y(x)=0 \] ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.138812 (sec), leaf count = 73

\[\left \{\left \{y(x)\to c_2 G_{1,2}^{2,0}\left (-\frac {1}{2 x^2}|\begin {array}{c} \frac {3}{2} \\ 0,0 \\\end {array}\right )+\frac {c_1 e^{\frac {1}{4 x^2}} \left (\left (2 x^2-1\right ) I_0\left (\frac {1}{4 x^2}\right )+I_1\left (\frac {1}{4 x^2}\right )\right )}{2 x^2}\right \}\right \}\]

Maple ✓
cpu = 0.115 (sec), leaf count = 85

\[ \left \{ y \left ( x \right ) ={\frac {{\it \_C1}}{{x}^{2}}{{\rm e}^{{\frac {1}{4\,{x}^{2}}}}} \left ( 2\,{{\sl I}_{0}\left (1/4\,{x}^{-2}\right )}{x}^{2}-{{\sl I}_{0}\left ({\frac {1}{4\,{x}^{2}}}\right )}+{{\sl I}_{1}\left ({\frac {1}{4\,{x}^{2}}}\right )} \right ) }+{\frac {{\it \_C2}}{{x}^{2}}{{\rm e}^{{\frac {1}{4\,{x}^{2}}}}} \left ( 2\,{{\sl K}_{0}\left (-1/4\,{x}^{-2}\right )}{x}^{2}-{{\sl K}_{0}\left (-{\frac {1}{4\,{x}^{2}}}\right )}+{{\sl K}_{1}\left (-{\frac {1}{4\,{x}^{2}}}\right )} \right ) } \right \} \] Mathematica raw input

DSolve[y[x] + x*(1 + x^2)*y'[x] + x^4*y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> (E^(1/(4*x^2))*((-1 + 2*x^2)*BesselI[0, 1/(4*x^2)] + BesselI[1, 1/(4*x
^2)])*C[1])/(2*x^2) + C[2]*MeijerG[{{}, {3/2}}, {{0, 0}, {}}, -1/(2*x^2)]}}

Maple raw input

dsolve(x^4*diff(diff(y(x),x),x)+x*(x^2+1)*diff(y(x),x)+y(x) = 0, y(x),'implicit')

Maple raw output

y(x) = _C1*exp(1/4/x^2)*(2*BesselI(0,1/4/x^2)*x^2-BesselI(0,1/4/x^2)+BesselI(1,1
/4/x^2))/x^2+_C2*exp(1/4/x^2)*(2*BesselK(0,-1/4/x^2)*x^2-BesselK(0,-1/4/x^2)+Bes
selK(1,-1/4/x^2))/x^2