32.8k views
5 votes
Use the laplace transform to solve the following differential equation: ′′ −2′ −15=1, (0)= 0, ′(0)=0

User Gvs Akhil
by
8.3k points

1 Answer

4 votes

To solve the given differential equation using the Laplace transform, we will follow these steps:

Step 1: Take the Laplace transform of both sides of the equation.

Step 2: Solve for the Laplace transform of the unknown function.

Step 3: Use the inverse Laplace transform to obtain the solution in the time domain.

Let's begin with Step 1:

Taking the Laplace transform of the given differential equation, we have:

s^2 * Y(s) - 2s * y(0) - y'(0) - 15Y(s) = 1/s

Here, Y(s) represents the Laplace transform of the unknown function y(t).

Now, applying the initial conditions y(0) = 0 and y'(0) = 0, we get:

s^2 * Y(s) - 15Y(s) = 1/s

Step 2:

To solve for Y(s), we can factor out Y(s) as a common factor:

Y(s) * (s^2 - 15) = 1/s

Dividing both sides by (s^2 - 15), we have:

Y(s) = 1 / (s * (s^2 - 15))

Now, we need to express the right side in partial fractions. Let's decompose it as follows:

1 / (s * (s^2 - 15)) = A/s + (Bs + C) / (s^2 - 15)

To determine the constants A, B, and C, we multiply both sides by the common denominator:

1 = A * (s^2 - 15) + (Bs + C) * s

Expanding and collecting like terms:

1 = (A * s^2 + Bs^2 + Cs) - 15A

Comparing coefficients of like powers of s:

0s^2: B = 0

1s: C = 0

s^2: A = -1/15

Therefore, the partial fraction decomposition is:

1 / (s * (s^2 - 15)) = -1 / (15s) + 0 / (s^2 - 15)

Substituting the partial fraction decomposition into Y(s), we get:

Y(s) = -1 / (15s) + 0 / (s^2 - 15)

Simplifying:

Y(s) = -1 / (15s)

Step 3:

Now, we need to find the inverse Laplace transform of Y(s) to obtain the solution in the time domain.

Using a standard Laplace transform table, we find that the inverse Laplace transform of -1 / (15s) is:

y(t) = -1/15 * (1 - e^(0t))

Since e^(0t) is equal to 1, we can simplify the equation further:

y(t) = -1/15 * (1 - 1)

y(t) = 0

Therefore, the solution to the given differential equation is y(t) = 0.

User BIOHAZARD
by
8.2k points