[ Not really an answer, but refinements needing more than the 500 characters of a comment ]

calc314's answer *is* an explicit solution :

```
import sympy
y=function("y")(x)
E1=diff(y(x),x,3)(x)-3*diff(y(x),x,2)(x)+diff(y(x),x)(x)-5*y(x)==0
Sol=sympy.dsolve(*map(sympy.sympify, [E1,y(x)]))._sage_()
Sol
y(x) == C3*e^(x*complex_root_of(x^3 - 3*x^2 + x - 5, 2)) + C2*e^(x*complex_root_of(x^3 - 3*x^2 + x - 5, 1)) + C1*e^(x*complex_root_of(x^3 - 3*x^2 + x - 5, 0))
```

The function `complex_root_of(P,i)`

is a formal function representing the `i`

th root of the polynomial `P`

(in a stable fashion), which can be evaluated numerically). See its help docstring, as well as `sympy`

's `CRoot`

's.

If symbolic solutions are needed, you can also extract the characteristic polynomial and solve it symbolically :

```
Sol.rhs().operands()[0].operands()[1].operands()[0].operands()[1].operands()[0].solve(x)
[x == -1/2*(1/9*sqrt(235)*sqrt(3) + 3)^(1/3)*(I*sqrt(3) + 1) + 1/3*(I*sqrt(3) - 1)/(1/9*sqrt(235)*sqrt(3) + 3)^(1/3) + 1,
x == -1/2*(1/9*sqrt(235)*sqrt(3) + 3)^(1/3)*(-I*sqrt(3) + 1) + 1/3*(-I*sqrt(3) - 1)/(1/9*sqrt(235)*sqrt(3) + 3)^(1/3) + 1,
x == (1/9*sqrt(235)*sqrt(3) + 3)^(1/3) + 2/3/(1/9*sqrt(235)*sqrt(3) + 3)^(1/3) + 1]
```

But I think that the "formal root" form s a better expression in a lot of situations.

giac gives an answer, but this answer fails to be converted to sage