QuestionQ(10) Find the general solution of the ODE .
Studdy Solution
STEP 1
1. The given differential equation is a linear homogeneous differential equation with constant coefficients.
2. The solution involves finding the characteristic equation and solving for the roots.
STEP 2
1. Write the characteristic equation.
2. Solve the characteristic equation for its roots.
3. Write the general solution based on the roots.
STEP 3
First, express the differential equation in terms of its characteristic equation. Assume a solution of the form , which implies:
Factor out (since ):
STEP 4
Factor the characteristic equation:
This gives us the roots:
1. implies (with multiplicity 3).
2. implies .
Thus, the roots are .
STEP 5
Write the general solution based on the roots. For the root with multiplicity 3, the solution is . For the roots and , the solutions are and , respectively.
Thus, the general solution is:
The general solution of the ODE is:
Was this helpful?