The solution involves finding the homogeneous solution to the differential equation with characteristic roots, then finding a particular solution to satisfy the non-homogeneous term, and finally applying the initial conditions to determine the specific solution.
- The initial value problem presented is a second-order linear differential equation with non-homogeneous term e-t.
- To find the solution that satisfies the condition y(t) → 0 as t → ∞, we can use the method of undetermined coefficients to find a particular solution and the method of characteristic equation to find the homogeneous solution.
- Firstly, we solve the homogeneous equation y'' + 36y = 0.
- The characteristic equation is r2 + 36 = 0, which has roots r = ±6i.
- This gives us a general homogeneous solution of yh(t) = A × cos(6t) + B × sin(6t).
- Secondly, for the non-homogeneous part, we guess a particular solution of form ke-t and find the value of k that satisfies the equation.
- After finding k, the particular solution yp(t) can be added to the homogeneous solution.
- The initial conditions y(0) = y0 and y'(0) = y'0 are used to solve for the constants A and B in the solution.
- By setting t = 0 in both the solution and its derivative and using the given initial conditions, we can find the values of A and B.
- Since y(t) → 0 as t → ∞, this indicates that the particular solution has a dominant effect at large t, which guides the values of A and B.