Laplace transforms worksheet with answers

Free printable Laplace transforms worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

New worksheet Practise online with hints

For teachers: free to print, copy and use in class. Every new worksheet has new values and its own answer key. A link to ExamProblems on your course page is appreciated.

Name: ____________________ Date: ____________

1.Easy

A signal f(t) has the Laplace transform F(s) = 12(s + 6)/(s(s + 1)(s + 2)). Use the final value theorem to find the value f(∞) that f(t) settles to.

F(s) = 12(s + 6)/(s(s + 1)(s + 2))

f(∞) =

2.Easy

A signal f(t) has the Laplace transform F(s) = 11(s + 5)/(s(s + 1)(s + 8)). Use the final value theorem to find the value f(∞) that f(t) settles to.

F(s) = 11(s + 5)/(s(s + 1)(s + 8))

f(∞) =

3.Medium

F(s) = (6s + 3)/(s² + 11s + 24). Its inverse transform has the form f(t) = A·e^−3t + B·e^−8t. Find the coefficient A by partial fractions.

F(s) = (6s + 3)/(s² + 11s + 24)

A =

4.Medium

F(s) = 30/(s² + 14s + 48). Its inverse transform has the form f(t) = A·e^−6t + B·e^−8t. Find the coefficient A by partial fractions.

F(s) = 30/(s² + 14s + 48)

A =

5.Hard

A system obeys y″ + 7y′ + 10y = 30·u(t). It starts at rest (y(0) = 0, y′(0) = 0) and the input is a step u(t) = 1 for t ≥ 0. Use the Laplace transform to find the output y(0.25).

y″ + 7y′ + 10y = 30·u(t)u(t) = 1y(0) = 0, y′(0) = 0t = 0.25 s

y(0.25) =

6.Hard

A system is described by the differential equation y″ + 4y′ + 13y = 4u′ + 10u, with input u(t) and output y(t). Find its transfer function G(s) = Y(s)/U(s) (zero initial conditions) and evaluate the impulse response g(t) at t = 0.3 s. Find g(0.3).

y″ + 4y′ + 13y = 4u′ + 10ut = 0.3 s

g(0.3) =

Answer key

1. f(∞) = 36
2. f(∞) = 6.875
3. A = −3
4. A = 15
5. y(0.25) = 0.5404
6. g(0.3) = 1.651

Full step-by-step solutions: practise this topic on examproblems.com.