[9:52:53 PM] Pardeep Narang: y''+2ty'−4y=8 y(0)=4 y'(0)=0 i need to solve this using laplace transforms
To solve the given differential equation using Laplace transforms, we need to perform the following steps:
Step 1: Take the Laplace transform of both sides of the equation.
Step 2: Solve the resulting algebraic equation for the Laplace transform of the unknown function.
Step 3: Take the inverse Laplace transform to find the solution in the time domain.
Let's go through these steps one by one.
Step 1: Take the Laplace transform of both sides of the equation.
Taking the Laplace transform of the given differential equation y'' + 2ty' - 4y = 8, we have:
s^2Y(s) - sy(0) - y'(0) + 2sY(s) - 2y(0) - 4Y(s) = 8 / (s^2)
where Y(s) represents the Laplace transform of the unknown function y(t), y(0) represents the initial value of y(t) at t = 0, and y'(0) represents the initial value of the derivative of y(t) at t = 0.
Step 2: Solve the resulting algebraic equation for the Laplace transform of the unknown function.
Rearranging the equation obtained from step 1, we get:
Y(s) * (s^2 + 2s - 4) - sy(0) - y'(0) - 2y(0) = 8 / (s^2)
Simplifying further and substituting the initial conditions y(0) = 4 and y'(0) = 0, we can write:
Y(s) * (s^2 + 2s - 4) - 4s - 8 = 8 / (s^2)
Y(s) * (s^2 + 2s - 4) = (8 + 4s + 8) / (s^2)
Y(s) * (s^2 + 2s - 4) = (12 + 4s) / (s^2)
Dividing both sides by (s^2 + 2s - 4), we get:
Y(s) = (12 + 4s) / [(s^2) * (s + 2)]
Step 3: Take the inverse Laplace transform to find the solution in the time domain.
Now, we apply partial fraction decomposition to the right side of the equation to obtain the inverse Laplace transform.
(12 + 4s) / [(s^2) * (s + 2)] = A / (s^2) + B / s + C / (s + 2)
Multiplying through by the common denominator and equating the coefficients, we find:
12 + 4s = A * s * (s + 2) + B * (s^2) + C * s
Solving this equation for A, B, and C, we get:
A = 3, B = -6, C = 1
Substituting these values back into the partial fraction decomposition, we have:
(12 + 4s) / [(s^2) * (s + 2)] = (3 / (s^2)) - (6 / s) + (1 / (s + 2))
Taking the inverse Laplace transform of both sides, we obtain the final solution in the time domain:
y(t) = 3t - 6 + e^(-2t)
Therefore, the solution to the given differential equation y'' + 2ty' - 4y = 8, with initial conditions y(0) = 4 and y'(0) = 0, is y(t) = 3t - 6 + e^(-2t).
To solve the given ordinary differential equation (ODE) using Laplace transforms, we need to follow these steps:
Step 1: Take the Laplace transform of both sides of the equation.
Step 2: Use the initial conditions to simplify the equation.
Step 3: Solve for the Laplace transform of the dependent variable.
Step 4: Partial fraction decomposition (if required).
Step 5: Inverse Laplace transform to obtain the solution to the original equation.
Now, let's go through each step in detail.
Step 1: Take the Laplace transform of both sides of the equation.
We have the ODE:
y'' + 2ty' - 4y = 8
Taking the Laplace transform, we get:
L(y'') + 2tL(y') - 4L(y) = 8L(1)
Using the derivative property of Laplace transforms, we have:
s^2Y(s) - sy(0) - y'(0) + 2sY(s) - 2y(0) - 4Y(s) = 8/s
This can be rewritten as:
(s^2 + 2s - 4)Y(s) - s(4) - 0 + 2s(4) - 2(4) = 8/s
Simplifying further:
(s^2 + 2s - 4)Y(s) - 12s - 16 = 8/s
Step 2: Use the initial conditions to simplify the equation.
Given initial conditions are:
y(0) = 4
y'(0) = 0
Using these initial conditions, we can substitute the values:
(0^2 + 2(0) - 4)Y(s) - 12(0) - 16 = 8/s
This simplifies to:
-4Y(s) - 16 = 8/s
Step 3: Solve for the Laplace transform of the dependent variable.
Rearranging the equation:
-4Y(s) = 8/s + 16
Dividing through by -4:
Y(s) = -2/s - 4
Step 4: Partial fraction decomposition (if required).
The Laplace transform of (-2/s - 4) can be further simplified using partial fraction decomposition. However, in this case, it is already in its simplest form.
Step 5: Inverse Laplace transform to obtain the solution to the original equation.
Now, taking the inverse Laplace transform of Y(s) = -2/s - 4, we can find the solution to the original ODE:
y(t) = -2 - 4t
Therefore, the solution to the given ODE, using Laplace transforms, is y(t) = -2 - 4t.