Tutorial 9: Laplace Transform 2
Question PDF
- Evaluate the given Laplace transform. i.
ii.
iii.
iv.
v.
vi.
Solution
Question (i)
Question (ii)
Question (iii)
Question (iv)
Question (v)
Question (vi)
- Use the Laplace transform to solve the initial-value problem.
i.
ii. where
iii.
iv.
v.
Solution
Question (i)
Let
Question (ii)
The Laplace transform of the differential equation is
Question (iii)
The Laplace transform of the differential equation is
Question (iv)
The Laplace transform of the differential equation is
Let
Let
Question (v)
- Use the Laplace transform to solve the given system of differential equations.
i.
ii.
iii.
Solution
Question (i)
From , and subsitute into
By partial fraction,
Therefore,
Question (ii)
Let
By partial fraction,
Therefore,
Question (iii)
Then,
- Two masses and are connected to three springs of negligible mass having spring constants and , respectively.
Let and represent displacements of masses and from their equilibrium positions. The motion of the coupled system is represented by the system of second-order differential equations:
Using Laplace transform to solve the system when and .
Solution
For
Substitute unknowns:
Laplace transform:
For
Substitute unknowns:
Laplace transform:
Hence,
From
Substitute into
Substitute into
Inverse Laplace,
- The system of differential equations for the charge on the capacitor and the current in the electrical network shown below is
Find the charge on the capacitor using Laplace transform when .
Solution
Solving and :