If the function is
[tex]y = 4 e^x \cos(3x)[/tex]
then we have derivatives
[tex]Dy = 4 e^x \cos(3x) - 12 e^x \sin(3x)[/tex]
[tex]D^2y = -32 e^x \cos(3x) - 24 e^x \sin(3x)[/tex]
Now consider the linear ODE
[tex]aD^2y + bDy + cy = 0[/tex]
Substituting [tex]y[/tex] and its derivatives reduces the equation to
[tex](-32a + 4b + 4c) \cos(3x) + (-24a - 12b) \sin(3x) = 0[/tex]
Now,
[tex]-24a - 12b = 0 \implies b = -2a[/tex]
[tex]-32a + 4b + 4c = 0 \implies c = 10a[/tex]
Then the minimal ODE with the given solution is
[tex]aD^2y -2a Dy + 10ay = 0[/tex]
or
[tex]\boxed{D^2y -2 Dy + 10y = 0}[/tex]
