Question

The solution to the Initial value problem x ′′ +2x ′ +65x=4cos(8t),x(0)=0,x ′ (0)=0 is the sum of the steady periodic solution x sp​ and the transient solution x tr​ . Find both x sp​ and xtr​

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Answer 1

To solve the given initial value problem, we need to find the complementary function and the particular integral separately. The complementary function is found by solving the auxiliary equation, while the particular integral is found by assuming a solution of a certain form and solving for the coefficients. The general solution is the sum of the complementary function and the particular integral.

Step-by-step explanation:

The given initial value problem represents a second-order linear homogeneous differential equation with a constant coefficient. To solve this problem, we can find the complementary function (transient solution) and the particular integral (steady periodic solution) separately.

The complementary function can be found by solving the auxiliary equation: r^2 + 2r + 65 = 0. By factoring or using the quadratic formula, we can find the roots r1 and r2. The complementary function is then given by x_c(t) = c1e^(r1t) + c2e^(r2t).

The particular integral can be found by assuming a solution of the form x_p(t) = A*cos(8t) + B*sin(8t). Substituting this into the differential equation, we can equate the coefficients of the cos(8t) and sin(8t) terms to solve for A and B. Finally, the general solution is the sum of the complementary function and the particular integral: x(t) = x_c(t) + x_p(t).