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Laplace and Inverse Laplace Transforms: Solved Problems
Published on May 23, 2026
Calculus
Step-by-step solutions for Laplace transform and inverse Laplace transform problems involving exponential, cosine, and hyperbolic functions.
Question
Answer
Here are the step-by-step solutions for the Laplace transform and inverse Laplace transform problems shown in the image.
Basic Formulas Used:
Solutions:
1.
Using the formula
Using the formula
2.
Using the inverse formula
Using the inverse formula
3.
Note: This is a non-standard form. Usually, it’s
Assuming the function is
Note: This is a non-standard form. Usually, it’s
Assuming the function is
4.
Using the formula
Using the formula
5.
We solve term by term:
We solve term by term:
-
-
(since ) -
Result:
6.
First, rewrite the denominator to match standard forms:
Result:
First, rewrite the denominator to match standard forms:
Result:
7.
Multiply and divide by
Result:
Multiply and divide by
Result:
8.
Complete the square in the denominator:
Using the shifting theorem
Result:
Complete the square in the denominator:
Using the shifting theorem
Result:
9.
Using
Result:
Using
Result:
10.
(This is identical to problem 9)
Result:
(This is identical to problem 9)
Result: