4.46.10 \(x^2 (a+b+c+3) y'''(x)+x^3 y''''(x)+x (a b+a c+a+b c+b+c+1) y''(x)-y'(x) (x-a b c)-k y(x)=0\)

ODE
\[ x^2 (a+b+c+3) y'''(x)+x^3 y''''(x)+x (a b+a c+a+b c+b+c+1) y''(x)-y'(x) (x-a b c)-k y(x)=0 \] ODE Classification

[[_high_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.208939 (sec), leaf count = 146

\[\left \{\left \{y(x)\to x \left ((-1)^{1-a} c_2 x^{-a} \, _1F_3(-a+k+1;2-a,-a+b+1,-a+c+1;x)+(-1)^{1-b} c_3 x^{-b} \, _1F_3(-b+k+1;2-b,a-b+1,-b+c+1;x)+(-1)^{1-c} c_4 x^{-c} \, _1F_3(-c+k+1;2-c,a-c+1,b-c+1;x)\right )+c_1 \, _1F_3(k;a,b,c;x)\right \}\right \}\]

Maple ✓
cpu = 0.07 (sec), leaf count = 123

\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{\mbox {$_1$F$_3$}(k;\,a,b,c;\,x)}+{\it \_C2}\,{x}^{1-a}{\mbox {$_1$F$_3$}(1-a+k;\,2-a,1-a+c,1-a+b;\,x)}+{\it \_C3}\,{x}^{1-b}{\mbox {$_1$F$_3$}(1-b+k;\,2-b,1+a-b,1-b+c;\,x)}+{\it \_C4}\,{x}^{-c+1}{\mbox {$_1$F$_3$}(1-c+k;\,2-c,1-c+b,1-c+a;\,x)} \right \} \] Mathematica raw input

DSolve[-(k*y[x]) - (-(a*b*c) + x)*y'[x] + (1 + a + b + a*b + c + a*c + b*c)*x*y''[x] + (3 + a + b + c)*x^2*y'''[x] + x^3*y''''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> C[1]*HypergeometricPFQ[{k}, {a, b, c}, x] + x*(((-1)^(1 - a)*C[2]*Hype
rgeometricPFQ[{1 - a + k}, {2 - a, 1 - a + b, 1 - a + c}, x])/x^a + ((-1)^(1 - b
)*C[3]*HypergeometricPFQ[{1 - b + k}, {2 - b, 1 + a - b, 1 - b + c}, x])/x^b + (
(-1)^(1 - c)*C[4]*HypergeometricPFQ[{1 - c + k}, {2 - c, 1 + a - c, 1 + b - c}, 
x])/x^c)}}

Maple raw input

dsolve(x^3*diff(diff(diff(diff(y(x),x),x),x),x)+(3+a+b+c)*x^2*diff(diff(diff(y(x),x),x),x)+(a*b+a*c+b*c+a+b+c+1)*x*diff(diff(y(x),x),x)-(-a*b*c+x)*diff(y(x),x)-k*y(x) = 0, y(x),'implicit')

Maple raw output

y(x) = _C1*hypergeom([k],[a, b, c],x)+_C2*x^(1-a)*hypergeom([1-a+k],[2-a, 1-a+c,
 1-a+b],x)+_C3*x^(1-b)*hypergeom([1-b+k],[2-b, 1+a-b, 1-b+c],x)+_C4*x^(-c+1)*hyp
ergeom([1-c+k],[2-c, 1-c+b, 1-c+a],x)