\[ -f(x)+x^2 y^{(3)}(x)+\left (x^2+2\right ) y'(x)+4 x y''(x)+3 x y(x)=0 \] ✓ Mathematica : cpu = 1.01232 (sec), leaf count = 2585
DSolve[-f[x] + 3*x*y[x] + (2 + x^2)*Derivative[1][y][x] + 4*x*Derivative[2][y][x] + x^2*Derivative[3][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to J_0(x) c_1+2 Y_0(x) c_2+\frac {2 c_3 \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {x^2}{4}\right )}{x}+\frac {x J_0(x) \int _1^x\left (\frac {-16 J_1(K[1]) Y_0(K[1]){}^2 f(K[1]) \, _1F_2\left (3;\frac {5}{2},\frac {5}{2};-\frac {1}{4} K[1]^2\right ) K[1]^3+16 J_0(K[1]) Y_0(K[1]) Y_1(K[1]) f(K[1]) \, _1F_2\left (3;\frac {5}{2},\frac {5}{2};-\frac {1}{4} K[1]^2\right ) K[1]^3+18 J_2(K[1]) Y_0(K[1]){}^2 f(K[1]) \, _1F_2\left (2;\frac {3}{2},\frac {3}{2};-\frac {1}{4} K[1]^2\right ) K[1]^2-18 J_0(K[1]) Y_0(K[1]) Y_2(K[1]) f(K[1]) \, _1F_2\left (2;\frac {3}{2},\frac {3}{2};-\frac {1}{4} K[1]^2\right ) K[1]^2-9 J_1(K[1]) Y_0(K[1]){}^2 f(K[1]) \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[1]^2\right ) K[1]+9 J_0(K[1]) Y_0(K[1]) Y_1(K[1]) f(K[1]) \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[1]^2\right ) K[1]-9 J_2(K[1]) Y_0(K[1]) Y_1(K[1]) f(K[1]) \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[1]^2\right ) K[1]+9 J_1(K[1]) Y_0(K[1]) Y_2(K[1]) f(K[1]) \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[1]^2\right ) K[1]+36 J_1(K[1]) Y_0(K[1]){}^2 f(K[1]) \, _1F_2\left (2;\frac {3}{2},\frac {3}{2};-\frac {1}{4} K[1]^2\right ) K[1]-36 J_0(K[1]) Y_0(K[1]) Y_1(K[1]) f(K[1]) \, _1F_2\left (2;\frac {3}{2},\frac {3}{2};-\frac {1}{4} K[1]^2\right ) K[1]+9 J_2(K[1]) Y_0(K[1]){}^2 f(K[1]) \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[1]^2\right )+36 J_0(K[1]) Y_1(K[1]){}^2 f(K[1]) \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[1]^2\right )-36 J_1(K[1]) Y_0(K[1]) Y_1(K[1]) f(K[1]) \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[1]^2\right )-9 J_0(K[1]) Y_0(K[1]) Y_2(K[1]) f(K[1]) \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[1]^2\right )}{2 (J_0(K[1]) Y_1(K[1])-J_1(K[1]) Y_0(K[1])) \left (16 J_1(K[1]) Y_0(K[1]) \, _1F_2\left (3;\frac {5}{2},\frac {5}{2};-\frac {1}{4} K[1]^2\right ) K[1]^4-16 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]^4-18 J_2(K[1]) Y_0(K[1]) \, _1F_2\left (2;\frac {3}{2},\frac {3}{2};-\frac {1}{4} K[1]^2\right ) K[1]^3+18 J_0(K[1]) Y_2(K[1]) \, _1F_2\left (2;\frac {3}{2},\frac {3}{2};-\frac {1}{4} K[1]^2\right ) K[1]^3+9 J_1(K[1]) Y_0(K[1]) \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[1]^2\right ) K[1]^2-9 J_0(K[1]) Y_1(K[1]) \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[1]^2\right ) K[1]^2+9 J_2(K[1]) Y_1(K[1]) \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[1]^2\right ) K[1]^2-9 J_1(K[1]) Y_2(K[1]) \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[1]^2\right ) K[1]^2+36 J_1(K[1]) Y_0(K[1]) \, _1F_2\left (2;\frac {3}{2},\frac {3}{2};-\frac {1}{4} K[1]^2\right ) K[1]^2-36 J_0(K[1]) Y_1(K[1]) \, _1F_2\left (2;\frac {3}{2},\frac {3}{2};-\frac {1}{4} K[1]^2\right ) K[1]^2-9 J_2(K[1]) Y_0(K[1]) \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[1]^2\right ) K[1]+9 J_0(K[1]) Y_2(K[1]) \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[1]^2\right ) K[1]+36 J_1(K[1]) Y_0(K[1]) \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[1]^2\right )-36 J_0(K[1]) Y_1(K[1]) \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[1]^2\right )\right )}-\frac {Y_0(K[1]) f(K[1])}{2 (J_1(K[1]) Y_0(K[1])-J_0(K[1]) Y_1(K[1])) K[1]}\right )dK[1]+2 x Y_0(x) \int _1^x\left (\frac {16 J_0(K[2]) J_1(K[2]) Y_0(K[2]) f(K[2]) \, _1F_2\left (3;\frac {5}{2},\frac {5}{2};-\frac {1}{4} K[2]^2\right ) K[2]^3-16 J_0(K[2]){}^2 Y_1(K[2]) f(K[2]) \, _1F_2\left (3;\frac {5}{2},\frac {5}{2};-\frac {1}{4} K[2]^2\right ) K[2]^3-18 J_0(K[2]) J_2(K[2]) Y_0(K[2]) f(K[2]) \, _1F_2\left (2;\frac {3}{2},\frac {3}{2};-\frac {1}{4} K[2]^2\right ) K[2]^2+18 J_0(K[2]){}^2 Y_2(K[2]) f(K[2]) \, _1F_2\left (2;\frac {3}{2},\frac {3}{2};-\frac {1}{4} K[2]^2\right ) K[2]^2+9 J_0(K[2]) J_1(K[2]) Y_0(K[2]) f(K[2]) \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[2]^2\right ) K[2]-9 J_0(K[2]){}^2 Y_1(K[2]) f(K[2]) \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[2]^2\right ) K[2]+9 J_0(K[2]) J_2(K[2]) Y_1(K[2]) f(K[2]) \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[2]^2\right ) K[2]-9 J_0(K[2]) J_1(K[2]) Y_2(K[2]) f(K[2]) \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[2]^2\right ) K[2]-36 J_0(K[2]) J_1(K[2]) Y_0(K[2]) f(K[2]) \, _1F_2\left (2;\frac {3}{2},\frac {3}{2};-\frac {1}{4} K[2]^2\right ) K[2]+36 J_0(K[2]){}^2 Y_1(K[2]) f(K[2]) \, _1F_2\left (2;\frac {3}{2},\frac {3}{2};-\frac {1}{4} K[2]^2\right ) K[2]+36 J_1(K[2]){}^2 Y_0(K[2]) f(K[2]) \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[2]^2\right )-9 J_0(K[2]) J_2(K[2]) Y_0(K[2]) f(K[2]) \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[2]^2\right )-36 J_0(K[2]) J_1(K[2]) Y_1(K[2]) f(K[2]) \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[2]^2\right )+9 J_0(K[2]){}^2 Y_2(K[2]) f(K[2]) \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[2]^2\right )}{4 (J_1(K[2]) Y_0(K[2])-J_0(K[2]) Y_1(K[2])) \left (-16 J_1(K[2]) Y_0(K[2]) \, _1F_2\left (3;\frac {5}{2},\frac {5}{2};-\frac {1}{4} K[2]^2\right ) K[2]^4+16 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+18 J_2(K[2]) Y_0(K[2]) \, _1F_2\left (2;\frac {3}{2},\frac {3}{2};-\frac {1}{4} K[2]^2\right ) K[2]^3-18 J_0(K[2]) Y_2(K[2]) \, _1F_2\left (2;\frac {3}{2},\frac {3}{2};-\frac {1}{4} K[2]^2\right ) K[2]^3-9 J_1(K[2]) Y_0(K[2]) \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[2]^2\right ) K[2]^2+9 J_0(K[2]) Y_1(K[2]) \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[2]^2\right ) K[2]^2-9 J_2(K[2]) Y_1(K[2]) \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[2]^2\right ) K[2]^2+9 J_1(K[2]) Y_2(K[2]) \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[2]^2\right ) K[2]^2-36 J_1(K[2]) Y_0(K[2]) \, _1F_2\left (2;\frac {3}{2},\frac {3}{2};-\frac {1}{4} K[2]^2\right ) K[2]^2+36 J_0(K[2]) Y_1(K[2]) \, _1F_2\left (2;\frac {3}{2},\frac {3}{2};-\frac {1}{4} K[2]^2\right ) K[2]^2+9 J_2(K[2]) Y_0(K[2]) \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[2]^2\right ) K[2]-9 J_0(K[2]) Y_2(K[2]) \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[2]^2\right ) K[2]-36 J_1(K[2]) Y_0(K[2]) \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[2]^2\right )+36 J_0(K[2]) Y_1(K[2]) \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[2]^2\right )\right )}-\frac {J_0(K[2]) f(K[2])}{4 (J_0(K[2]) Y_1(K[2])-J_1(K[2]) Y_0(K[2])) K[2]}\right )dK[2]+2 \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {x^2}{4}\right ) \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]}{16 J_1(K[3]) Y_0(K[3]) \, _1F_2\left (3;\frac {5}{2},\frac {5}{2};-\frac {1}{4} K[3]^2\right ) K[3]^4-16 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]^4-18 J_2(K[3]) Y_0(K[3]) \, _1F_2\left (2;\frac {3}{2},\frac {3}{2};-\frac {1}{4} K[3]^2\right ) K[3]^3+18 J_0(K[3]) Y_2(K[3]) \, _1F_2\left (2;\frac {3}{2},\frac {3}{2};-\frac {1}{4} K[3]^2\right ) K[3]^3+9 J_1(K[3]) Y_0(K[3]) \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[3]^2\right ) K[3]^2-9 J_0(K[3]) Y_1(K[3]) \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[3]^2\right ) K[3]^2+9 J_2(K[3]) Y_1(K[3]) \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[3]^2\right ) K[3]^2-9 J_1(K[3]) Y_2(K[3]) \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[3]^2\right ) K[3]^2+36 J_1(K[3]) Y_0(K[3]) \, _1F_2\left (2;\frac {3}{2},\frac {3}{2};-\frac {1}{4} K[3]^2\right ) K[3]^2-36 J_0(K[3]) Y_1(K[3]) \, _1F_2\left (2;\frac {3}{2},\frac {3}{2};-\frac {1}{4} K[3]^2\right ) K[3]^2-9 J_2(K[3]) Y_0(K[3]) \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[3]^2\right ) K[3]+9 J_0(K[3]) Y_2(K[3]) \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[3]^2\right ) K[3]+36 J_1(K[3]) Y_0(K[3]) \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[3]^2\right )-36 J_0(K[3]) Y_1(K[3]) \, _1F_2\left (1;\frac {1}{2},\frac {1}{2};-\frac {1}{4} K[3]^2\right )}dK[3]}{x}\right \}\right \}\] ✓ Maple : cpu = 0.3 (sec), leaf count = 1033
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)=0,y(x))
\[\text {Expression too large to display}\]