Solve (D^2+6D+9)y=25sint

Solve it using ODE and C.F AND PI

To solve the differential equation \((D^2 + 6D + 9)y = 25\sin(t)\), we will break it down into two main parts: finding the complementary function (C.F.) and the particular integral (P.I.). The operator \(D\) represents differentiation with respect to \(t\), so \(D^2\) is the second derivative, and \(D\) is the first derivative. Let's dive into the solution step by step.

Step 1: Finding the Complementary Function (C.F.)

The first step is to solve the homogeneous equation associated with the given differential equation:

(D^2 + 6D + 9)y = 0

We can rewrite this as:

D^2 + 6D + 9 = 0

Next, we find the characteristic equation:

r^2 + 6r + 9 = 0

This can be factored as:

(r + 3)^2 = 0

This gives us a repeated root:

r = -3

For a repeated root, the complementary function is given by:

C.F. = C_1 e^{-3t} + C_2 t e^{-3t}

Step 2: Finding the Particular Integral (P.I.)

Now, we need to find a particular integral for the non-homogeneous part \(25\sin(t)\). Since the right-hand side is a sine function, we can use the method of undetermined coefficients. We will assume a solution of the form:

P.I. = A\sin(t) + B\cos(t)

Next, we differentiate this assumed solution:

P.I.' = A\cos(t) - B\sin(t)

P.I.'' = -A\sin(t) - B\cos(t)

Now, we substitute \(P.I.\), \(P.I.'\), and \(P.I.''\) back into the left-hand side of the original equation:

(D^2 + 6D + 9)(A\sin(t) + B\cos(t)) = 25\sin(t)

Calculating this gives:

(-A\sin(t) - B\cos(t)) + 6(A\cos(t) - B\sin(t)) + 9(A\sin(t) + B\cos(t)) = 25\sin(t)

Combining like terms, we get:

[(9A - A - 6B)\sin(t) + (9B - B + 6A)\cos(t)] = 25\sin(t)

This simplifies to:

(8A - 6B)\sin(t) + (8B + 6A)\cos(t) = 25\sin(t)

From here, we can set up a system of equations:

• 8A - 6B = 25
• 8B + 6A = 0

Now, we can solve this system. From the second equation, we can express \(B\) in terms of \(A\):

B = -\frac{6}{8}A = -\frac{3}{4}A

Substituting this into the first equation:

8A - 6(-\frac{3}{4}A) = 25

This simplifies to:

8A + \frac{18}{4}A = 25

Combining terms gives:

(8 + 4.5)A = 25

12.5A = 25

Thus, we find:

A = 2

Now substituting \(A\) back to find \(B\):

B = -\frac{3}{4}(2) = -\frac{3}{2}

So, the particular integral is:

P.I. = 2\sin(t) - \frac{3}{2}\cos(t)

Step 3: General Solution

The general solution \(y(t)\) is the sum of the complementary function and the particular integral:

y(t) = C_1 e^{-3t} + C_2 t e^{-3t} + 2\sin(t) - \frac{3}{2}\cos(t)

And there you have it! This is the complete solution to the differential equation \((D^2 + 6D + 9)y = 25\sin(t)\). If you have any questions about any of the steps, feel free to ask!