**Assignment 2 Question**

**Question PDF**

**Deadline**10 June 2022, 12 noon,

**by hand**

The heat released by a certain radioactive substance upon nuclear fission can be described by the following second-order linear nonhomogeneous differential equation:

where is the heat released in Joule, is the time in microseconds and is the last digit of your matrix number. For those whose matrix number ending 0, you should use . You are required to solve the equation analytically:

- Perform the Laplace transform of the above equation and express in its simplest term. The initial conditions are given as and .
**(40 marks)**

- By performing an inverse Laplace transform based on your answer (a), express the amount of heat released () as a function of time ().
**(20 marks)**

- A second additional effect arises from a sudden rapid but short release of heat amounting to Joule at microseconds. Rewrite the second order differential equation.
**(10 marks)**

- Solve the equation in (c) by using the Laplace transform technique. The initial conditions are the same as (a). Hint: You may apply the superposition principle.
**(30 marks)**

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#### Solution

**Question (a)**

*where*

*is the last digit of matrix number.*

Performing Laplace transform,

**(10 marks)**

Given and

By partial fraction,

**(10 marks)**

Solving Eq (Hint: Gaussian Elimination / Direct Substitution),

**(10 marks)**

**(10 marks)**

**Question (b)**

Performing inverse Laplace Transform:

**(20 marks)**

**Question (c)**

**(10 marks)**

**Question (d)**

Applying superposition principle based on the second term on the RHS:

Applying Laplace Transform,

**(10 marks)**

Performing Inverse Laplace Transform,

**(10 marks)**

Combining the solution from Question (c),

**(10 marks)**