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1. Find the inverse Laplace transform of X(S)=-5/(s+3) (s-2)?
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1. Find the inverse Laplace transform of X(S)=-5/(s+3) (s-2)?
Understanding the Given Function
To find the inverse Laplace transform of X(S) = -5 / ((s + 3)(s - 2)), we first need to decompose this expression into simpler fractions.
Partial Fraction Decomposition
1. Setup the Decomposition:
- We express X(S) as A/(s + 3) + B/(s - 2).
2. Multiply Through:
- -5 = A(s - 2) + B(s + 3).
3. Solve for A and B:
- Choose convenient values for s to solve for A and B:
- For s = 2: -5 = A(2 - 2) + B(2 + 3) => B = -1.
- For s = -3: -5 = A(-3 - 2) + B(-3 + 3) => A = 1.
- Thus, X(S) = 1/(s + 3) - 1/(s - 2).
Inverse Laplace Transform
1. Apply Inverse Transform:
- The inverse Laplace transforms of the components are known:
- L^-1{1/(s + 3)} = e^(-3t).
- L^-1{-1/(s - 2)} = -e^(2t).
2. Combine Results:
- Therefore, the inverse Laplace transform of X(S) is:
- L^-1{X(S)} = e^(-3t) - e^(2t).
Final Result
The final expression for the inverse Laplace transform of X(S) = -5 / ((s + 3)(s - 2)) is:
L^-1{X(S)} = e^(-3t) - e^(2t).
This function represents the time-domain response corresponding to the given system in the Laplace domain.
To find the inverse Laplace transform of X(S) = -5 / ((s + 3)(s - 2)), we first need to decompose this expression into simpler fractions.
Partial Fraction Decomposition
1. Setup the Decomposition:
- We express X(S) as A/(s + 3) + B/(s - 2).
2. Multiply Through:
- -5 = A(s - 2) + B(s + 3).
3. Solve for A and B:
- Choose convenient values for s to solve for A and B:
- For s = 2: -5 = A(2 - 2) + B(2 + 3) => B = -1.
- For s = -3: -5 = A(-3 - 2) + B(-3 + 3) => A = 1.
- Thus, X(S) = 1/(s + 3) - 1/(s - 2).
Inverse Laplace Transform
1. Apply Inverse Transform:
- The inverse Laplace transforms of the components are known:
- L^-1{1/(s + 3)} = e^(-3t).
- L^-1{-1/(s - 2)} = -e^(2t).
2. Combine Results:
- Therefore, the inverse Laplace transform of X(S) is:
- L^-1{X(S)} = e^(-3t) - e^(2t).
Final Result
The final expression for the inverse Laplace transform of X(S) = -5 / ((s + 3)(s - 2)) is:
L^-1{X(S)} = e^(-3t) - e^(2t).
This function represents the time-domain response corresponding to the given system in the Laplace domain.
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1. Find the inverse Laplace transform of X(S)=-5/(s+3) (s-2)? for Electronics and Communication Engineering (ECE) 2024 is part of Electronics and Communication Engineering (ECE) preparation. The Question and answers have been prepared according to the Electronics and Communication Engineering (ECE) exam syllabus. Information about 1. Find the inverse Laplace transform of X(S)=-5/(s+3) (s-2)? covers all topics & solutions for Electronics and Communication Engineering (ECE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 1. Find the inverse Laplace transform of X(S)=-5/(s+3) (s-2)?.
1. Find the inverse Laplace transform of X(S)=-5/(s+3) (s-2)? for Electronics and Communication Engineering (ECE) 2024 is part of Electronics and Communication Engineering (ECE) preparation. The Question and answers have been prepared according to the Electronics and Communication Engineering (ECE) exam syllabus. Information about 1. Find the inverse Laplace transform of X(S)=-5/(s+3) (s-2)? covers all topics & solutions for Electronics and Communication Engineering (ECE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 1. Find the inverse Laplace transform of X(S)=-5/(s+3) (s-2)?.
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