The characteristic equation of this ODE has real and distinct roots: 4, 1, and 7. Hence the solution is composed entirely of exponential functions.
The characteristic equation of this ODE has two pairs of equal roots: and . The repeated roots give rise to the basis of the solutions, .
The characteristic equation for this ODE has two pairs of roots with nonzero imaginary parts: , , , and . Hence the solution basis can be expressed with trigonometric and exponential functions.