Question

Solving a second order Differential equation using Laplace transform :
A. Partial fraction expansion
B. Apply initial conditions in Laplace domain
C. Take Laplace transform of differential equation
D. Obtain time domain solution Choose the correct answer from the options given below :

• A. A, B, C, D
• B. A, C, B, D
• C. C, B, A, D
• D. A, D, C, B

Hint:

Standard Laplace transform method for differential equations: \[ \text{Transform} \rightarrow \text{Apply initial conditions} \rightarrow \text{Partial fractions} \rightarrow \text{Inverse transform} \]

Answer 1

The Correct Option is C

Concept: Laplace Transform is an extremely powerful method for solving differential equations, especially initial value problems. The standard procedure involves:
• Taking Laplace transform of the differential equation,
• Applying initial conditions,
• Simplifying the transformed equation,
• Using partial fractions if necessary,
• Taking inverse Laplace transform to return to time domain. The sequence of operations is very important.

Step 1: Understanding step \(C\). Statement \(C\): \[ \text{Take Laplace transform of differential equation} \] This is always the first step because the differential equation must first be converted into an algebraic equation in the Laplace domain.

Step 2: Understanding step \(B\). Statement \(B\): \[ \text{Apply initial conditions in Laplace domain} \] After taking the Laplace transform, derivatives produce terms involving initial conditions. For example: \[ \mathcal{L}\left(\frac{dy}{dt}\right)=sY(s)-y(0) \] Thus initial conditions are substituted immediately after transforming.

Step 3: Understanding step \(A\). Statement \(A\): \[ \text{Partial fraction expansion} \] After simplification, the transformed solution \(Y(s)\) is often expressed as rational fractions. Partial fractions are used to simplify inverse Laplace transformation.

Step 4: Understanding step \(D\). Statement \(D\): \[ \text{Obtain time domain solution} \] Finally inverse Laplace transform is taken to return from \(s\)-domain to the original time-domain solution.

Step 5: Arranging the correct order. Thus the proper sequence is: \[ C \to B \to A \to D \] Therefore: \[ \boxed{C,B,A,D} \] Hence the correct option is: \[ \boxed{(3)} \]