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 = 1.37434 (sec), leaf count = 871

\[\left \{\left \{y(x)\to 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]^4+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]))) K[1]^2\right )+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]^4+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]))) K[2]^2\right )+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]+\frac {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]^4+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]))) K[3]^2\right )+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 = 1.96 (sec), leaf count = 1849

\[\text {Expression too large to display}\] 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] -> BesselJ[0, x]*C[1] + 2*BesselY[0, x]*C[2] + BesselJ[0, x]*Inactive[Int
egrate][(-18*f[K[1]]*(2*BesselY[0, K[1]]*HypergeometricPFQ[{2}, {3/2, 3/2}, -1/4
*K[1]^2]*K[1]^2 + HypergeometricPFQ[{1}, {1/2, 1/2}, -1/4*K[1]^2]*(BesselY[0, K[
1]] - BesselY[1, K[1]]*K[1])))/(K[1]*(2*(8*(BesselJ[1, K[1]]*BesselY[0, K[1]] -
BesselJ[0, K[1]]*BesselY[1, K[1]])*HypergeometricPFQ[{3}, {5/2, 5/2}, -1/4*K[1]^
2]*K[1]^4 + 9*HypergeometricPFQ[{2}, {3/2, 3/2}, -1/4*K[1]^2]*K[1]^2*(2*BesselJ[
1, K[1]]*BesselY[0, K[1]] - BesselJ[2, K[1]]*BesselY[0, K[1]]*K[1] + BesselJ[0,
K[1]]*(-2*BesselY[1, K[1]] + BesselY[2, K[1]]*K[1]))) + 9*HypergeometricPFQ[{1},
{1/2, 1/2}, -1/4*K[1]^2]*(BesselJ[2, K[1]]*K[1]*(-BesselY[0, K[1]] + BesselY[1,
K[1]]*K[1]) + BesselJ[1, K[1]]*(-(BesselY[2, K[1]]*K[1]^2) + BesselY[0, K[1]]*(
4 + K[1]^2)) + BesselJ[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]*Inactive[Integrate][(9*f[K[2]]*(2*Be
sselJ[0, K[2]]*HypergeometricPFQ[{2}, {3/2, 3/2}, -1/4*K[2]^2]*K[2]^2 + Hypergeo
metricPFQ[{1}, {1/2, 1/2}, -1/4*K[2]^2]*(BesselJ[0, K[2]] - BesselJ[1, K[2]]*K[2
])))/(K[2]*(2*(8*(BesselJ[1, K[2]]*BesselY[0, K[2]] - BesselJ[0, K[2]]*BesselY[1
, K[2]])*HypergeometricPFQ[{3}, {5/2, 5/2}, -1/4*K[2]^2]*K[2]^4 + 9*Hypergeometr
icPFQ[{2}, {3/2, 3/2}, -1/4*K[2]^2]*K[2]^2*(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*HypergeometricPFQ[{1}, {1/2, 1/2}, -1/4*K[2]^2]*
(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] - BesselY[1, K[2]]*(4 + K[2]^2))))), {K[2], 1, x}]
+ (2*HypergeometricPFQ[{1}, {1/2, 1/2}, -1/4*x^2]*(C[3] + Inactive[Integrate][(9
*(BesselJ[1, K[3]]*BesselY[0, K[3]] - BesselJ[0, K[3]]*BesselY[1, K[3]])*f[K[3]]
*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}, -1/4*K[3]^2]*K[3]^4 + 9*HypergeometricPF
Q[{2}, {3/2, 3/2}, -1/4*K[3]^2]*K[3]^2*(2*BesselJ[1, K[3]]*BesselY[0, K[3]] - Be
sselJ[2, K[3]]*BesselY[0, K[3]]*K[3] + BesselJ[0, K[3]]*(-2*BesselY[1, K[3]] + B
esselY[2, K[3]]*K[3]))) + 9*HypergeometricPFQ[{1}, {1/2, 1/2}, -1/4*K[3]^2]*(Bes
selJ[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))

Maple raw output

[y(x) = (Int(9*(hypergeom([2],[3/2, 3/2],-1/4*x^2)*MeijerG([[-1/2], []],[[0, 0,
-1/2], []],1/4*x^2)*x^2+MeijerG([[-3/2], []],[[0, 0, -1/2], []],1/4*x^2)*hyperge
om([1],[1/2, 1/2],-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))*f(x)/(-4*hypergeom([3],[5/2, 5/2],-1/4*x^2)*Bes
selJ(1,x)*MeijerG([[-1/2], []],[[0, 0, -1/2], []],1/4*x^2)*x^5-9*BesselJ(0,x)*hy
pergeom([2],[3/2, 3/2],-1/4*x^2)*MeijerG([[-1/2], []],[[0, 0, -1/2], []],1/4*x^2
)*x^4-8*BesselJ(0,x)*MeijerG([[-3/2], []],[[0, 0, -1/2], []],1/4*x^2)*hypergeom(
[3],[5/2, 5/2],-1/4*x^2)*x^4+12*BesselJ(0,x)*hypergeom([3],[5/2, 5/2],-1/4*x^2)*
MeijerG([[-1/2], []],[[0, 0, -1/2], []],1/4*x^2)*x^4+144*BesselJ(0,x)*hypergeom(
[2],[3/2, 3/2],-1/4*x^2)*MeijerG([[-3/2], []],[[0, 0, -1/2], []],1/4*x^2)*x^2-36
*BesselJ(0,x)*hypergeom([2],[3/2, 3/2],-1/4*x^2)*MeijerG([[-5/2], []],[[0, 0, -1
/2], []],1/4*x^2)*x^2-81*BesselJ(0,x)*hypergeom([2],[3/2, 3/2],-1/4*x^2)*MeijerG
([[-1/2], []],[[0, 0, -1/2], []],1/4*x^2)*x^2-9*BesselJ(0,x)*MeijerG([[-3/2], []
],[[0, 0, -1/2], []],1/4*x^2)*hypergeom([1],[1/2, 1/2],-1/4*x^2)*x^2+9*BesselJ(0
,x)*hypergeom([1],[1/2, 1/2],-1/4*x^2)*MeijerG([[-1/2], []],[[0, 0, -1/2], []],1
/4*x^2)*x^2-72*MeijerG([[-3/2], []],[[0, 0, -1/2], []],1/4*x^2)*BesselJ(1,x)*hyp
ergeom([1],[1/2, 1/2],-1/4*x^2)*x+18*MeijerG([[-5/2], []],[[0, 0, -1/2], []],1/4
*x^2)*BesselJ(1,x)*hypergeom([1],[1/2, 1/2],-1/4*x^2)*x+36*BesselJ(1,x)*hypergeo
m([1],[1/2, 1/2],-1/4*x^2)*MeijerG([[-1/2], []],[[0, 0, -1/2], []],1/4*x^2)*x+63
*BesselJ(0,x)*MeijerG([[-3/2], []],[[0, 0, -1/2], []],1/4*x^2)*hypergeom([1],[1/
2, 1/2],-1/4*x^2)-18*BesselJ(0,x)*MeijerG([[-5/2], []],[[0, 0, -1/2], []],1/4*x^
2)*hypergeom([1],[1/2, 1/2],-1/4*x^2)-27*BesselJ(0,x)*hypergeom([1],[1/2, 1/2],-
1/4*x^2)*MeijerG([[-1/2], []],[[0, 0, -1/2], []],1/4*x^2)),x)*BesselJ(0,x)*x-Int
(9/2*(2*BesselJ(0,x)*hypergeom([2],[3/2, 3/2],-1/4*x^2)*x^2-BesselJ(1,x)*hyperge
om([1],[1/2, 1/2],-1/4*x^2)*x+BesselJ(0,x)*hypergeom([1],[1/2, 1/2],-1/4*x^2))*f
(x)/(-4*hypergeom([3],[5/2, 5/2],-1/4*x^2)*BesselJ(1,x)*MeijerG([[-1/2], []],[[0
, 0, -1/2], []],1/4*x^2)*x^5-9*BesselJ(0,x)*hypergeom([2],[3/2, 3/2],-1/4*x^2)*M
eijerG([[-1/2], []],[[0, 0, -1/2], []],1/4*x^2)*x^4-8*BesselJ(0,x)*MeijerG([[-3/
2], []],[[0, 0, -1/2], []],1/4*x^2)*hypergeom([3],[5/2, 5/2],-1/4*x^2)*x^4+12*Be
sselJ(0,x)*hypergeom([3],[5/2, 5/2],-1/4*x^2)*MeijerG([[-1/2], []],[[0, 0, -1/2]
, []],1/4*x^2)*x^4+144*BesselJ(0,x)*hypergeom([2],[3/2, 3/2],-1/4*x^2)*MeijerG([
[-3/2], []],[[0, 0, -1/2], []],1/4*x^2)*x^2-36*BesselJ(0,x)*hypergeom([2],[3/2,
3/2],-1/4*x^2)*MeijerG([[-5/2], []],[[0, 0, -1/2], []],1/4*x^2)*x^2-81*BesselJ(0
,x)*hypergeom([2],[3/2, 3/2],-1/4*x^2)*MeijerG([[-1/2], []],[[0, 0, -1/2], []],1
/4*x^2)*x^2-9*BesselJ(0,x)*MeijerG([[-3/2], []],[[0, 0, -1/2], []],1/4*x^2)*hype
rgeom([1],[1/2, 1/2],-1/4*x^2)*x^2+9*BesselJ(0,x)*hypergeom([1],[1/2, 1/2],-1/4*
x^2)*MeijerG([[-1/2], []],[[0, 0, -1/2], []],1/4*x^2)*x^2-72*MeijerG([[-3/2], []
],[[0, 0, -1/2], []],1/4*x^2)*BesselJ(1,x)*hypergeom([1],[1/2, 1/2],-1/4*x^2)*x+
18*MeijerG([[-5/2], []],[[0, 0, -1/2], []],1/4*x^2)*BesselJ(1,x)*hypergeom([1],[
1/2, 1/2],-1/4*x^2)*x+36*BesselJ(1,x)*hypergeom([1],[1/2, 1/2],-1/4*x^2)*MeijerG
([[-1/2], []],[[0, 0, -1/2], []],1/4*x^2)*x+63*BesselJ(0,x)*MeijerG([[-3/2], []]
,[[0, 0, -1/2], []],1/4*x^2)*hypergeom([1],[1/2, 1/2],-1/4*x^2)-18*BesselJ(0,x)*
MeijerG([[-5/2], []],[[0, 0, -1/2], []],1/4*x^2)*hypergeom([1],[1/2, 1/2],-1/4*x
^2)-27*BesselJ(0,x)*hypergeom([1],[1/2, 1/2],-1/4*x^2)*MeijerG([[-1/2], []],[[0,
0, -1/2], []],1/4*x^2)),x)*MeijerG([[-1/2], []],[[0, 0, -1/2], []],1/4*x^2)*x-I
nt(9/2*(BesselJ(1,x)*MeijerG([[-1/2], []],[[0, 0, -1/2], []],1/4*x^2)*x+2*Bessel
J(0,x)*MeijerG([[-3/2], []],[[0, 0, -1/2], []],1/4*x^2)-3*BesselJ(0,x)*MeijerG([
[-1/2], []],[[0, 0, -1/2], []],1/4*x^2))*x*f(x)/(-4*hypergeom([3],[5/2, 5/2],-1/
4*x^2)*BesselJ(1,x)*MeijerG([[-1/2], []],[[0, 0, -1/2], []],1/4*x^2)*x^5-9*Besse
lJ(0,x)*hypergeom([2],[3/2, 3/2],-1/4*x^2)*MeijerG([[-1/2], []],[[0, 0, -1/2], [
]],1/4*x^2)*x^4-8*BesselJ(0,x)*MeijerG([[-3/2], []],[[0, 0, -1/2], []],1/4*x^2)*
hypergeom([3],[5/2, 5/2],-1/4*x^2)*x^4+12*BesselJ(0,x)*hypergeom([3],[5/2, 5/2],
-1/4*x^2)*MeijerG([[-1/2], []],[[0, 0, -1/2], []],1/4*x^2)*x^4+144*BesselJ(0,x)*
hypergeom([2],[3/2, 3/2],-1/4*x^2)*MeijerG([[-3/2], []],[[0, 0, -1/2], []],1/4*x
^2)*x^2-36*BesselJ(0,x)*hypergeom([2],[3/2, 3/2],-1/4*x^2)*MeijerG([[-5/2], []],
[[0, 0, -1/2], []],1/4*x^2)*x^2-81*BesselJ(0,x)*hypergeom([2],[3/2, 3/2],-1/4*x^
2)*MeijerG([[-1/2], []],[[0, 0, -1/2], []],1/4*x^2)*x^2-9*BesselJ(0,x)*MeijerG([
[-3/2], []],[[0, 0, -1/2], []],1/4*x^2)*hypergeom([1],[1/2, 1/2],-1/4*x^2)*x^2+9
*BesselJ(0,x)*hypergeom([1],[1/2, 1/2],-1/4*x^2)*MeijerG([[-1/2], []],[[0, 0, -1
/2], []],1/4*x^2)*x^2-72*MeijerG([[-3/2], []],[[0, 0, -1/2], []],1/4*x^2)*Bessel
J(1,x)*hypergeom([1],[1/2, 1/2],-1/4*x^2)*x+18*MeijerG([[-5/2], []],[[0, 0, -1/2
], []],1/4*x^2)*BesselJ(1,x)*hypergeom([1],[1/2, 1/2],-1/4*x^2)*x+36*BesselJ(1,x
)*hypergeom([1],[1/2, 1/2],-1/4*x^2)*MeijerG([[-1/2], []],[[0, 0, -1/2], []],1/4
*x^2)*x+63*BesselJ(0,x)*MeijerG([[-3/2], []],[[0, 0, -1/2], []],1/4*x^2)*hyperge
om([1],[1/2, 1/2],-1/4*x^2)-18*BesselJ(0,x)*MeijerG([[-5/2], []],[[0, 0, -1/2],
[]],1/4*x^2)*hypergeom([1],[1/2, 1/2],-1/4*x^2)-27*BesselJ(0,x)*hypergeom([1],[1
/2, 1/2],-1/4*x^2)*MeijerG([[-1/2], []],[[0, 0, -1/2], []],1/4*x^2)),x)*hypergeo
m([1],[1/2, 1/2],-1/4*x^2))/x+_C1*BesselJ(0,x)+_C2*hypergeom([1],[1/2, 1/2],-1/4
*x^2)/x+_C3*MeijerG([[-1/2], []],[[0, 0, -1/2], []],1/4*x^2)]

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