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.