Question

Use The Laplace Transform To Solve The Given Initial Value Problem. Y′′−10y′−24y=0; Y(0)=9, Y′(0)=10

Y(T)= _______

Answer 1

To solve the given initial value problem using the Laplace transform, we need to find the Laplace transform of y''(t), y'(t), and y(t) and substitute them into the given differential equation. After finding the Laplace transforms, we can solve the resulting equation to find the solution y(t).

Step-by-step explanation:

To solve the given initial value problem using the Laplace transform, we need to find the Laplace transform of y''(t), y'(t), and y(t) and substitute them into the given differential equation.

Let's start by finding the Laplace transform of y''(t):

L{y''(t)} = s^2Y(s) - sy(0) - y'(0)

Substituting the given initial values, we get:

L{y''(t)} = s^2Y(s) - 9s - 10

Similarly, we can find the Laplace transforms of y'(t) and y(t):

L{y'(t)} = sY(s) - y(0) = sY(s) - 10

L{y(t)} = Y(s)

Substituting the Laplace transforms into the given differential equation, we have:

s^2Y(s) - 10sY(s) - 24Y(s) = 0

Factoring out Y(s), we get:

(s^2 - 10s - 24)Y(s) = 0

Now we can solve for Y(s) by setting the coefficient term equal to zero:

s^2 - 10s - 24 = 0

Using the quadratic formula, we find two possible values for s:

s = 12 or s = -2

Now we can find the inverse Laplace transform of Y(s) to get the solution y(t).