x2y′′′(x) + ( x2 + 2) y′(x) + 4xy′′(x) + 3xy(x) = f(x)

4.44.16 $x^2 y'''(x)+\left (x^2+2\right ) y'(x)+4 x y''(x)+3 x y(x)=f(x)$

ODE
\[ x^2 y'''(x)+\left (x^2+2\right ) y'(x)+4 x y''(x)+3 x y(x)=f(x) \] ODE Classiﬁcation

[[_3rd_order, _linear, _nonhomogeneous]]

Book solution method
TO DO

Mathematica ✓
cpu = 6.79624 (sec), leaf count = 872

\[\left \{\left \{y(x)\to \frac {x \left (J_0(x) c_1+2 Y_0(x) c_2+J_0(x) \int _1^x -\frac {18 f(K[1]) \left (2 Y_0(K[1]) \, _1F_2\left (2;\frac {3}{2},\frac {3}{2};-\frac {1}{4} K[1]^2\right ) K[1]^2+\, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[1]^2\right ) (Y_0(K[1])-Y_1(K[1]) K[1])\right )}{K[1] \left (2 \left (8 (J_1(K[1]) Y_0(K[1])-J_0(K[1]) Y_1(K[1])) \, _1F_2\left (3;\frac {5}{2},\frac {5}{2};-\frac {1}{4} K[1]^2\right ) K[1]^2+9 \, _1F_2\left (2;\frac {3}{2},\frac {3}{2};-\frac {1}{4} K[1]^2\right ) (2 J_1(K[1]) Y_0(K[1])-J_2(K[1]) K[1] Y_0(K[1])+J_0(K[1]) (Y_2(K[1]) K[1]-2 Y_1(K[1])))\right ) K[1]^2+9 \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[1]^2\right ) \left (J_2(K[1]) K[1] (Y_1(K[1]) K[1]-Y_0(K[1]))+J_1(K[1]) \left (Y_0(K[1]) \left (K[1]^2+4\right )-Y_2(K[1]) K[1]^2\right )+J_0(K[1]) \left (Y_2(K[1]) K[1]-Y_1(K[1]) \left (K[1]^2+4\right )\right )\right )\right )} \, dK[1]+2 Y_0(x) \int _1^x \frac {9 f(K[2]) \left (2 J_0(K[2]) \, _1F_2\left (2;\frac {3}{2},\frac {3}{2};-\frac {1}{4} K[2]^2\right ) K[2]^2+\, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[2]^2\right ) (J_0(K[2])-J_1(K[2]) K[2])\right )}{K[2] \left (2 \left (8 (J_1(K[2]) Y_0(K[2])-J_0(K[2]) Y_1(K[2])) \, _1F_2\left (3;\frac {5}{2},\frac {5}{2};-\frac {1}{4} K[2]^2\right ) K[2]^2+9 \, _1F_2\left (2;\frac {3}{2},\frac {3}{2};-\frac {1}{4} K[2]^2\right ) (2 J_1(K[2]) Y_0(K[2])-J_2(K[2]) K[2] Y_0(K[2])+J_0(K[2]) (Y_2(K[2]) K[2]-2 Y_1(K[2])))\right ) K[2]^2+9 \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[2]^2\right ) \left (J_2(K[2]) K[2] (Y_1(K[2]) K[2]-Y_0(K[2]))+J_1(K[2]) \left (Y_0(K[2]) \left (K[2]^2+4\right )-Y_2(K[2]) K[2]^2\right )+J_0(K[2]) \left (Y_2(K[2]) K[2]-Y_1(K[2]) \left (K[2]^2+4\right )\right )\right )\right )} \, dK[2]\right )+2 \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {x^2}{4}\right ) \left (c_3+\int _1^x \frac {9 (J_1(K[3]) Y_0(K[3])-J_0(K[3]) Y_1(K[3])) f(K[3]) K[3]}{2 \left (8 (J_1(K[3]) Y_0(K[3])-J_0(K[3]) Y_1(K[3])) \, _1F_2\left (3;\frac {5}{2},\frac {5}{2};-\frac {1}{4} K[3]^2\right ) K[3]^2+9 \, _1F_2\left (2;\frac {3}{2},\frac {3}{2};-\frac {1}{4} K[3]^2\right ) (2 J_1(K[3]) Y_0(K[3])-J_2(K[3]) K[3] Y_0(K[3])+J_0(K[3]) (Y_2(K[3]) K[3]-2 Y_1(K[3])))\right ) K[3]^2+9 \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[3]^2\right ) \left (J_2(K[3]) K[3] (Y_1(K[3]) K[3]-Y_0(K[3]))+J_1(K[3]) \left (Y_0(K[3]) \left (K[3]^2+4\right )-Y_2(K[3]) K[3]^2\right )+J_0(K[3]) \left (Y_2(K[3]) K[3]-Y_1(K[3]) \left (K[3]^2+4\right )\right )\right )} \, dK[3]\right )}{x}\right \}\right \}\]

Maple ✓
cpu = 0.34 (sec), leaf count = 1033

\[ \left \{ y \left ( x \right ) ={\frac {1}{x} \left ( \int \!18\,{\frac {f \left ( x \right ) \left ( \left ( -1/2\,x{{\sl J}_{1}\left (x\right )}+1/2\,{{\sl J}_{0}\left (x\right )} \right ) {\mbox {$_1$F$_2$}(1;\,1/2,1/2;\,-1/4\,{x}^{2})}+{\mbox {$_1$F$_2$}(2;\,3/2,3/2;\,-1/4\,{x}^{2})}{{\sl J}_{0}\left (x\right )}{x}^{2} \right ) }{ \left ( \left ( -18\,{x}^{2}{{\sl J}_{0}\left (x\right )}-72\,x{{\sl J}_{1}\left (x\right )}+54\,{{\sl J}_{0}\left (x\right )} \right ) {\mbox {$_1$F$_2$}(1;\,1/2,1/2;\,-1/4\,{x}^{2})}+8\, \left ( 9/4\,{{\sl J}_{0}\left (x\right )} \left ( {x}^{2}+9 \right ) {\mbox {$_1$F$_2$}(2;\,3/2,3/2;\,-1/4\,{x}^{2})}+{x}^{2}{\mbox {$_1$F$_2$}(3;\,5/2,5/2;\,-1/4\,{x}^{2})} \left ( x{{\sl J}_{1}\left (x\right )}-3\,{{\sl J}_{0}\left (x\right )} \right ) \right ) {x}^{2} \right ) G^{3, 1}_{1, 3}\left (1/4\,{x}^{2}\, \Big \vert \,^{-1/2}_{0, 0, -1/2}\right )+ \left ( \left ( 18\,{x}^{2}{{\sl J}_{0}\left (x\right )}+144\,x{{\sl J}_{1}\left (x\right )}-126\,{{\sl J}_{0}\left (x\right )} \right ) {\mbox {$_1$F$_2$}(1;\,1/2,1/2;\,-1/4\,{x}^{2})}+16\,{x}^{2}{{\sl J}_{0}\left (x\right )} \left ( {x}^{2}{\mbox {$_1$F$_2$}(3;\,5/2,5/2;\,-1/4\,{x}^{2})}-18\,{\mbox {$_1$F$_2$}(2;\,3/2,3/2;\,-1/4\,{x}^{2})} \right ) \right ) G^{3, 1}_{1, 3}\left (1/4\,{x}^{2}\, \Big \vert \,^{-3/2}_{0, 0, -1/2}\right )+72\,G^{3, 1}_{1, 3}\left (1/4\,{x}^{2}\, \Big \vert \,^{-5/2}_{0, 0, -1/2}\right ) \left ( \left ( -1/2\,x{{\sl J}_{1}\left (x\right )}+1/2\,{{\sl J}_{0}\left (x\right )} \right ) {\mbox {$_1$F$_2$}(1;\,1/2,1/2;\,-1/4\,{x}^{2})}+{\mbox {$_1$F$_2$}(2;\,3/2,3/2;\,-1/4\,{x}^{2})}{{\sl J}_{0}\left (x\right )}{x}^{2} \right ) }}\,{\rm d}xG^{3, 1}_{1, 3}\left ({\frac {{x}^{2}}{4}}\, \Big \vert \,^{-{\frac {1}{2}}}_{0, 0, -{\frac {1}{2}}}\right )x+{\it \_C3}\,G^{3, 1}_{1, 3}\left ({\frac {{x}^{2}}{4}}\, \Big \vert \,^{-{\frac {1}{2}}}_{0, 0, -{\frac {1}{2}}}\right )x-\int \!9\,{\frac {f \left ( x \right ) \left ( \left ( {x}^{2}{\mbox {$_1$F$_2$}(2;\,3/2,3/2;\,-1/4\,{x}^{2})}-{\mbox {$_1$F$_2$}(1;\,1/2,1/2;\,-1/4\,{x}^{2})} \right ) G^{3, 1}_{1, 3}\left (1/4\,{x}^{2}\, \Big \vert \,^{-1/2}_{0, 0, -1/2}\right )+{\mbox {$_1$F$_2$}(1;\,1/2,1/2;\,-1/4\,{x}^{2})}G^{3, 1}_{1, 3}\left (1/4\,{x}^{2}\, \Big \vert \,^{-3/2}_{0, 0, -1/2}\right ) \right ) }{ \left ( \left ( -9\,{x}^{2}{{\sl J}_{0}\left (x\right )}-36\,x{{\sl J}_{1}\left (x\right )}+27\,{{\sl J}_{0}\left (x\right )} \right ) {\mbox {$_1$F$_2$}(1;\,1/2,1/2;\,-1/4\,{x}^{2})}+4\, \left ( 9/4\,{{\sl J}_{0}\left (x\right )} \left ( {x}^{2}+9 \right ) {\mbox {$_1$F$_2$}(2;\,3/2,3/2;\,-1/4\,{x}^{2})}+{x}^{2}{\mbox {$_1$F$_2$}(3;\,5/2,5/2;\,-1/4\,{x}^{2})} \left ( x{{\sl J}_{1}\left (x\right )}-3\,{{\sl J}_{0}\left (x\right )} \right ) \right ) {x}^{2} \right ) G^{3, 1}_{1, 3}\left (1/4\,{x}^{2}\, \Big \vert \,^{-1/2}_{0, 0, -1/2}\right )+ \left ( \left ( 9\,{x}^{2}{{\sl J}_{0}\left (x\right )}+72\,x{{\sl J}_{1}\left (x\right )}-63\,{{\sl J}_{0}\left (x\right )} \right ) {\mbox {$_1$F$_2$}(1;\,1/2,1/2;\,-1/4\,{x}^{2})}+8\,{x}^{2}{{\sl J}_{0}\left (x\right )} \left ( {x}^{2}{\mbox {$_1$F$_2$}(3;\,5/2,5/2;\,-1/4\,{x}^{2})}-18\,{\mbox {$_1$F$_2$}(2;\,3/2,3/2;\,-1/4\,{x}^{2})} \right ) \right ) G^{3, 1}_{1, 3}\left (1/4\,{x}^{2}\, \Big \vert \,^{-3/2}_{0, 0, -1/2}\right )+36\,G^{3, 1}_{1, 3}\left (1/4\,{x}^{2}\, \Big \vert \,^{-5/2}_{0, 0, -1/2}\right ) \left ( \left ( -1/2\,x{{\sl J}_{1}\left (x\right )}+1/2\,{{\sl J}_{0}\left (x\right )} \right ) {\mbox {$_1$F$_2$}(1;\,1/2,1/2;\,-1/4\,{x}^{2})}+{\mbox {$_1$F$_2$}(2;\,3/2,3/2;\,-1/4\,{x}^{2})}{{\sl J}_{0}\left (x\right )}{x}^{2} \right ) }}\,{\rm d}x{{\sl J}_{0}\left (x\right )}x+{\it \_C1}\,{{\sl J}_{0}\left (x\right )}x-\int \!-9\,{\frac {xf \left ( x \right ) \left ( \left ( x{{\sl J}_{1}\left (x\right )}-3\,{{\sl J}_{0}\left (x\right )} \right ) G^{3, 1}_{1, 3}\left (1/4\,{x}^{2}\, \Big \vert \,^{-1/2}_{0, 0, -1/2}\right )+2\,G^{3, 1}_{1, 3}\left (1/4\,{x}^{2}\, \Big \vert \,^{-3/2}_{0, 0, -1/2}\right ){{\sl J}_{0}\left (x\right )} \right ) }{ \left ( \left ( -18\,{x}^{2}{{\sl J}_{0}\left (x\right )}-72\,x{{\sl J}_{1}\left (x\right )}+54\,{{\sl J}_{0}\left (x\right )} \right ) {\mbox {$_1$F$_2$}(1;\,1/2,1/2;\,-1/4\,{x}^{2})}+8\, \left ( 9/4\,{{\sl J}_{0}\left (x\right )} \left ( {x}^{2}+9 \right ) {\mbox {$_1$F$_2$}(2;\,3/2,3/2;\,-1/4\,{x}^{2})}+{x}^{2}{\mbox {$_1$F$_2$}(3;\,5/2,5/2;\,-1/4\,{x}^{2})} \left ( x{{\sl J}_{1}\left (x\right )}-3\,{{\sl J}_{0}\left (x\right )} \right ) \right ) {x}^{2} \right ) G^{3, 1}_{1, 3}\left (1/4\,{x}^{2}\, \Big \vert \,^{-1/2}_{0, 0, -1/2}\right )+ \left ( \left ( 18\,{x}^{2}{{\sl J}_{0}\left (x\right )}+144\,x{{\sl J}_{1}\left (x\right )}-126\,{{\sl J}_{0}\left (x\right )} \right ) {\mbox {$_1$F$_2$}(1;\,1/2,1/2;\,-1/4\,{x}^{2})}+16\,{x}^{2}{{\sl J}_{0}\left (x\right )} \left ( {x}^{2}{\mbox {$_1$F$_2$}(3;\,5/2,5/2;\,-1/4\,{x}^{2})}-18\,{\mbox {$_1$F$_2$}(2;\,3/2,3/2;\,-1/4\,{x}^{2})} \right ) \right ) G^{3, 1}_{1, 3}\left (1/4\,{x}^{2}\, \Big \vert \,^{-3/2}_{0, 0, -1/2}\right )+72\,G^{3, 1}_{1, 3}\left (1/4\,{x}^{2}\, \Big \vert \,^{-5/2}_{0, 0, -1/2}\right ) \left ( \left ( -1/2\,x{{\sl J}_{1}\left (x\right )}+1/2\,{{\sl J}_{0}\left (x\right )} \right ) {\mbox {$_1$F$_2$}(1;\,1/2,1/2;\,-1/4\,{x}^{2})}+{\mbox {$_1$F$_2$}(2;\,3/2,3/2;\,-1/4\,{x}^{2})}{{\sl J}_{0}\left (x\right )}{x}^{2} \right ) }}\,{\rm d}x{\mbox {$_1$F$_2$}(1;\,{\frac {1}{2}},{\frac {1}{2}};\,-{\frac {{x}^{2}}{4}})}+{\it \_C2}\,{\mbox {$_1$F$_2$}(1;\,{\frac {1}{2}},{\frac {1}{2}};\,-{\frac {{x}^{2}}{4}})} \right ) } \right \} \] Mathematica raw input

DSolve[3*x*y[x] + (2 + x^2)*y'[x] + 4*x*y''[x] + x^2*y'''[x] == f[x],y[x],x]

Mathematica raw output

{{y[x] -> (x*(BesselJ[0, x]*C[1] + 2*BesselY[0, x]*C[2] + BesselJ[0, x]*Integrat
e[(-18*f[K[1]]*(2*BesselY[0, K[1]]*HypergeometricPFQ[{2}, {3/2, 3/2}, -K[1]^2/4]
*K[1]^2 + HypergeometricPFQ[{1}, {1/2, 1/2}, -K[1]^2/4]*(BesselY[0, K[1]] - Bess
elY[1, K[1]]*K[1])))/(K[1]*(2*K[1]^2*(8*(BesselJ[1, K[1]]*BesselY[0, K[1]] - Bes
selJ[0, K[1]]*BesselY[1, K[1]])*HypergeometricPFQ[{3}, {5/2, 5/2}, -K[1]^2/4]*K[
1]^2 + 9*HypergeometricPFQ[{2}, {3/2, 3/2}, -K[1]^2/4]*(2*BesselJ[1, K[1]]*Besse
lY[0, K[1]] - BesselJ[2, K[1]]*BesselY[0, K[1]]*K[1] + BesselJ[0, K[1]]*(-2*Bess
elY[1, K[1]] + BesselY[2, K[1]]*K[1]))) + 9*HypergeometricPFQ[{1}, {1/2, 1/2}, -
K[1]^2/4]*(BesselJ[2, K[1]]*K[1]*(-BesselY[0, K[1]] + BesselY[1, K[1]]*K[1]) + B
esselJ[1, K[1]]*(-(BesselY[2, K[1]]*K[1]^2) + BesselY[0, K[1]]*(4 + K[1]^2)) + B
esselJ[0, K[1]]*(BesselY[2, K[1]]*K[1] - BesselY[1, K[1]]*(4 + K[1]^2))))), {K[1
], 1, x}] + 2*BesselY[0, x]*Integrate[(9*f[K[2]]*(2*BesselJ[0, K[2]]*Hypergeomet
ricPFQ[{2}, {3/2, 3/2}, -K[2]^2/4]*K[2]^2 + HypergeometricPFQ[{1}, {1/2, 1/2}, -
K[2]^2/4]*(BesselJ[0, K[2]] - BesselJ[1, K[2]]*K[2])))/(K[2]*(2*K[2]^2*(8*(Besse
lJ[1, K[2]]*BesselY[0, K[2]] - BesselJ[0, K[2]]*BesselY[1, K[2]])*Hypergeometric
PFQ[{3}, {5/2, 5/2}, -K[2]^2/4]*K[2]^2 + 9*HypergeometricPFQ[{2}, {3/2, 3/2}, -K
[2]^2/4]*(2*BesselJ[1, K[2]]*BesselY[0, K[2]] - BesselJ[2, K[2]]*BesselY[0, K[2]
]*K[2] + BesselJ[0, K[2]]*(-2*BesselY[1, K[2]] + BesselY[2, K[2]]*K[2]))) + 9*Hy
pergeometricPFQ[{1}, {1/2, 1/2}, -K[2]^2/4]*(BesselJ[2, K[2]]*K[2]*(-BesselY[0,
K[2]] + BesselY[1, K[2]]*K[2]) + BesselJ[1, K[2]]*(-(BesselY[2, K[2]]*K[2]^2) +
BesselY[0, K[2]]*(4 + K[2]^2)) + BesselJ[0, K[2]]*(BesselY[2, K[2]]*K[2] - Besse
lY[1, K[2]]*(4 + K[2]^2))))), {K[2], 1, x}]) + 2*HypergeometricPFQ[{1}, {1/2, 1/
2}, -x^2/4]*(C[3] + Integrate[(9*(BesselJ[1, K[3]]*BesselY[0, K[3]] - BesselJ[0,
K[3]]*BesselY[1, K[3]])*f[K[3]]*K[3])/(2*K[3]^2*(8*(BesselJ[1, K[3]]*BesselY[0,
K[3]] - BesselJ[0, K[3]]*BesselY[1, K[3]])*HypergeometricPFQ[{3}, {5/2, 5/2}, -
K[3]^2/4]*K[3]^2 + 9*HypergeometricPFQ[{2}, {3/2, 3/2}, -K[3]^2/4]*(2*BesselJ[1,
K[3]]*BesselY[0, K[3]] - BesselJ[2, K[3]]*BesselY[0, K[3]]*K[3] + BesselJ[0, K[
3]]*(-2*BesselY[1, K[3]] + BesselY[2, K[3]]*K[3]))) + 9*HypergeometricPFQ[{1}, {
1/2, 1/2}, -K[3]^2/4]*(BesselJ[2, K[3]]*K[3]*(-BesselY[0, K[3]] + BesselY[1, K[3
]]*K[3]) + BesselJ[1, K[3]]*(-(BesselY[2, K[3]]*K[3]^2) + BesselY[0, K[3]]*(4 +
K[3]^2)) + BesselJ[0, K[3]]*(BesselY[2, K[3]]*K[3] - BesselY[1, K[3]]*(4 + K[3]^
2)))), {K[3], 1, x}]))/x}}

Maple raw input

dsolve(x^2*diff(diff(diff(y(x),x),x),x)+4*x*diff(diff(y(x),x),x)+(x^2+2)*diff(y(x),x)+3*x*y(x) = f(x), y(x),'implicit')

Maple raw output

y(x) = (Int(18*f(x)*((-1/2*x*BesselJ(1,x)+1/2*BesselJ(0,x))*hypergeom([1],[1/2,
1/2],-1/4*x^2)+hypergeom([2],[3/2, 3/2],-1/4*x^2)*BesselJ(0,x)*x^2)/(((-18*x^2*B
esselJ(0,x)-72*x*BesselJ(1,x)+54*BesselJ(0,x))*hypergeom([1],[1/2, 1/2],-1/4*x^2
)+8*(9/4*BesselJ(0,x)*(x^2+9)*hypergeom([2],[3/2, 3/2],-1/4*x^2)+x^2*hypergeom([
3],[5/2, 5/2],-1/4*x^2)*(x*BesselJ(1,x)-3*BesselJ(0,x)))*x^2)*MeijerG([[-1/2], [
]],[[0, 0, -1/2], []],1/4*x^2)+((18*x^2*BesselJ(0,x)+144*x*BesselJ(1,x)-126*Bess
elJ(0,x))*hypergeom([1],[1/2, 1/2],-1/4*x^2)+16*x^2*BesselJ(0,x)*(x^2*hypergeom(
[3],[5/2, 5/2],-1/4*x^2)-18*hypergeom([2],[3/2, 3/2],-1/4*x^2)))*MeijerG([[-3/2]
, []],[[0, 0, -1/2], []],1/4*x^2)+72*MeijerG([[-5/2], []],[[0, 0, -1/2], []],1/4
*x^2)*((-1/2*x*BesselJ(1,x)+1/2*BesselJ(0,x))*hypergeom([1],[1/2, 1/2],-1/4*x^2)
+hypergeom([2],[3/2, 3/2],-1/4*x^2)*BesselJ(0,x)*x^2)),x)*MeijerG([[-1/2], []],[
[0, 0, -1/2], []],1/4*x^2)*x+_C3*MeijerG([[-1/2], []],[[0, 0, -1/2], []],1/4*x^2
)*x-Int(9*f(x)*((x^2*hypergeom([2],[3/2, 3/2],-1/4*x^2)-hypergeom([1],[1/2, 1/2]
,-1/4*x^2))*MeijerG([[-1/2], []],[[0, 0, -1/2], []],1/4*x^2)+hypergeom([1],[1/2,
1/2],-1/4*x^2)*MeijerG([[-3/2], []],[[0, 0, -1/2], []],1/4*x^2))/(((-9*x^2*Bess
elJ(0,x)-36*x*BesselJ(1,x)+27*BesselJ(0,x))*hypergeom([1],[1/2, 1/2],-1/4*x^2)+4
*(9/4*BesselJ(0,x)*(x^2+9)*hypergeom([2],[3/2, 3/2],-1/4*x^2)+x^2*hypergeom([3],
[5/2, 5/2],-1/4*x^2)*(x*BesselJ(1,x)-3*BesselJ(0,x)))*x^2)*MeijerG([[-1/2], []],
[[0, 0, -1/2], []],1/4*x^2)+((9*x^2*BesselJ(0,x)+72*x*BesselJ(1,x)-63*BesselJ(0,
x))*hypergeom([1],[1/2, 1/2],-1/4*x^2)+8*x^2*BesselJ(0,x)*(x^2*hypergeom([3],[5/
2, 5/2],-1/4*x^2)-18*hypergeom([2],[3/2, 3/2],-1/4*x^2)))*MeijerG([[-3/2], []],[
[0, 0, -1/2], []],1/4*x^2)+36*MeijerG([[-5/2], []],[[0, 0, -1/2], []],1/4*x^2)*(
(-1/2*x*BesselJ(1,x)+1/2*BesselJ(0,x))*hypergeom([1],[1/2, 1/2],-1/4*x^2)+hyperg
eom([2],[3/2, 3/2],-1/4*x^2)*BesselJ(0,x)*x^2)),x)*BesselJ(0,x)*x+_C1*BesselJ(0,
x)*x-Int(-9*f(x)*x*((x*BesselJ(1,x)-3*BesselJ(0,x))*MeijerG([[-1/2], []],[[0, 0,
-1/2], []],1/4*x^2)+2*MeijerG([[-3/2], []],[[0, 0, -1/2], []],1/4*x^2)*BesselJ(
0,x))/(((-18*x^2*BesselJ(0,x)-72*x*BesselJ(1,x)+54*BesselJ(0,x))*hypergeom([1],[
1/2, 1/2],-1/4*x^2)+8*(9/4*BesselJ(0,x)*(x^2+9)*hypergeom([2],[3/2, 3/2],-1/4*x^
2)+x^2*hypergeom([3],[5/2, 5/2],-1/4*x^2)*(x*BesselJ(1,x)-3*BesselJ(0,x)))*x^2)*
MeijerG([[-1/2], []],[[0, 0, -1/2], []],1/4*x^2)+((18*x^2*BesselJ(0,x)+144*x*Bes
selJ(1,x)-126*BesselJ(0,x))*hypergeom([1],[1/2, 1/2],-1/4*x^2)+16*x^2*BesselJ(0,
x)*(x^2*hypergeom([3],[5/2, 5/2],-1/4*x^2)-18*hypergeom([2],[3/2, 3/2],-1/4*x^2)
))*MeijerG([[-3/2], []],[[0, 0, -1/2], []],1/4*x^2)+72*MeijerG([[-5/2], []],[[0,
0, -1/2], []],1/4*x^2)*((-1/2*x*BesselJ(1,x)+1/2*BesselJ(0,x))*hypergeom([1],[1
/2, 1/2],-1/4*x^2)+hypergeom([2],[3/2, 3/2],-1/4*x^2)*BesselJ(0,x)*x^2)),x)*hype
rgeom([1],[1/2, 1/2],-1/4*x^2)+_C2*hypergeom([1],[1/2, 1/2],-1/4*x^2))/x

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4.44.16 \(x^2 y'''(x)+\left (x^2+2\right ) y'(x)+4 x y''(x)+3 x y(x)=f(x)\)