solve dT/t=k(T-Tm(t)) using laplace transformation, where Tm=50+20U(t-h) and where U = unit step function.
To solve the given differential equation using Laplace transformation, we need to follow these steps:
Step 1: Take the Laplace transformation of both sides of the equation.
Step 2: Replace the derivatives with the corresponding Laplace variables.
Step 3: Solve the resulting algebraic equation for the Laplace transform of T(t).
Step 4: Use the inverse Laplace transform to obtain the solution in the time domain if needed.
Let's go through the steps in detail:
Step 1: Take the Laplace transformation of both sides of the equation:
L{dT/dt} = L{k(T - Tm(t))}
Where L{} represents the Laplace transformation.
Step 2: Replace the derivatives with the corresponding Laplace variables:
sL{T} - T(0) = k (L{T} - L{Tm(t)})
Where s is the Laplace variable and T(0) represents the initial condition of T(t), i.e., the value of T at t = 0.
Step 3: Solve the resulting algebraic equation for the Laplace transform of T(t):
sL{T} - T(0) = k (L{T} - L{Tm(t)})
sL{T} - kL{T} = k( L{Tm(t)} - T(0))
Factor out T in the left-hand side:
(s - k) L{T} = k( L{Tm(t)} - T(0))
Divide through by (s - k):
L{T} = k( L{Tm(t)} - T(0)) / (s - k)
Step 4: Use the inverse Laplace transform to obtain the solution in the time domain:
Apply the inverse Laplace transform to both sides of the equation to get the expression for T(t):
T(t) = L^-1 { k( L{Tm(t)} - T(0)) / (s - k) }
Now, given that Tm(t) = 50 + 20U(t-h), where U(t-h) is the unit step function, we can substitute it into the equation:
T(t) = L^-1 { k( L{50 + 20U(t-h)} - T(0)) / (s - k) }
By manipulating the Laplace transforms of U(t-h) and substituting their inverse transforms, you can simplify the equation further.