Solve the equation x"+9x=-35(t-三). x(0)=1, x'(0）=0

We can start by finding the characteristic equation, which is:

r^2 + 9 = 0

Solving for r, we get:

r = ±√(-9) = ±3i

Therefore, the general solution to the homogeneous equation x" + 9x = 0 is:

x(t) = c1cos(3t) + c2sin(3t)

To find a particular solution to the non-homogeneous equation x" + 9x = -35(t-三), we can use the method of undetermined coefficients. Since the right-hand side is a linear function of t, we can guess a particular solution of the form:

x(t) = At + B

Taking the derivatives, we get:

x'(t) = A
x"(t) = 0

Substituting into the original equation, we get:

0 + 9(At+B) = -35(t-三)

Simplifying and equating coefficients, we get:

9A = -35
9B = -35三

Solving for A and B, we get:

A = -35/9
B = (-35三)/9

Therefore, the general solution to the non-homogeneous equation is:

x(t) = c1cos(3t) + c2sin(3t) - (35/9)t + (-35三/9)

Using the initial conditions x(0) = 1 and x'(0) = 0, we can solve for the constants c1 and c2:

x(0) = 1 = c1cos(0) + c2sin(0) - (35/9)(0) + (-35三/9)
1 = c1

x'(0) = 0 = -3c1sin(0) + 3c2cos(0) - 35/9
0 = 3c2 - 35/9

Solving for c1 and c2, we get:

c1 = 1
c2 = 35/27

Therefore, the solution to the initial value problem is:

x(t) = cos(3t) + (35/27)sin(3t) - (35/9)t - (35三/9)