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Differential Equations · Transforms & Systems

Laplace Transforms — Practice Problems and Worked Solutions

Move differential equations into the s-domain.

The formulas you need here

Laplace transform of a derivative

L {f^{'}} = s F (s) - f (0)

What it says: Differentiation becomes multiplication by s, with the initial condition folded in.
When to reach for it: Initial-value problems, especially with discontinuous or impulsive forcing.
Watch out: The initial values are part of the transform. Dropping f(0) silently solves a different problem.

L {f^{''}} = s^{2} F (s) - s f (0) - f^{'} (0)

Each extra derivative adds one more initial-condition term.

17 worked problems

Every step is generated and checked by a computer algebra system.

1.
Find the Laplace transform.
$L {1} (s)$
Answer: $\frac{1}{s}$
Step-by-step solution
1. $L {f} = \int_{0}^{\infty} e^{- s x} f (x) d x$
2. $= \frac{1}{s}$
2.
Find the Laplace transform.
$L {x} (s)$
Answer: $\frac{1}{s ^{2}}$
Step-by-step solution
1. $L {f} = \int_{0}^{\infty} e^{- s x} f (x) d x$
2. $= \frac{1}{s ^{2}}$
3.
Find the Laplace transform.
$L {x^{2}} (s)$
Answer: $\frac{2}{s ^{3}}$
Step-by-step solution
1. $L {f} = \int_{0}^{\infty} e^{- s x} f (x) d x$
2. $= \frac{2}{s ^{3}}$
4.
Find the Laplace transform.
$L {e^{2 x}} (s)$
Answer: $\frac{1}{s - 2}$
Step-by-step solution
1. $L {f} = \int_{0}^{\infty} e^{- s x} f (x) d x$
2. $= \frac{1}{s - 2}$
5.
Find the Laplace transform.
$L {sin (3 x)} (s)$
Answer: $\frac{3}{s ^{2} + 9}$
Step-by-step solution
1. $L {f} = \int_{0}^{\infty} e^{- s x} f (x) d x$
2. $= \frac{3}{s ^{2} + 9}$
6.
Find the Laplace transform.
$L {cos (2 x)} (s)$
Answer: $\frac{s}{s ^{2} + 4}$
Step-by-step solution
1. $L {f} = \int_{0}^{\infty} e^{- s x} f (x) d x$
2. $= \frac{s}{s ^{2} + 4}$
7.
Find the Laplace transform.
$L {x^{3}} (s)$
Answer: $\frac{6}{s ^{4}}$
Step-by-step solution
1. $L {f} = \int_{0}^{\infty} e^{- s x} f (x) d x$
2. $= \frac{6}{s ^{4}}$
8.
Find the Laplace transform.
$L {e^{3 x}} (s)$
Answer: $\frac{1}{s - 3}$
Step-by-step solution
1. $L {f} = \int_{0}^{\infty} e^{- s x} f (x) d x$
2. $= \frac{1}{s - 3}$
9.
Find the Laplace transform.
$L {sin (x)} (s)$
Answer: $\frac{1}{s ^{2} + 1}$
Step-by-step solution
1. $L {f} = \int_{0}^{\infty} e^{- s x} f (x) d x$
2. $= \frac{1}{s ^{2} + 1}$
10.
Find the Laplace transform.
$L {cos (x)} (s)$
Answer: $\frac{s}{s ^{2} + 1}$
Step-by-step solution
1. $L {f} = \int_{0}^{\infty} e^{- s x} f (x) d x$
2. $= \frac{s}{s ^{2} + 1}$
11.
Find the Laplace transform.
$L {e^{- x}} (s)$
Answer: $\frac{1}{s + 1}$
Step-by-step solution
1. $L {f} = \int_{0}^{\infty} e^{- s x} f (x) d x$
2. $= \frac{1}{s + 1}$
12.
Find the Laplace transform.
$L {x^{4}} (s)$
Answer: $\frac{24}{s ^{5}}$
Step-by-step solution
1. $L {f} = \int_{0}^{\infty} e^{- s x} f (x) d x$
2. $= \frac{24}{s ^{5}}$
13.
Find the Laplace transform.
$L {sin (2 x)} (s)$
Answer: $\frac{2}{s ^{2} + 4}$
Step-by-step solution
1. $L {f} = \int_{0}^{\infty} e^{- s x} f (x) d x$
2. $= \frac{2}{s ^{2} + 4}$
14.
Find the Laplace transform.
$L {cos (3 x)} (s)$
Answer: $\frac{s}{s ^{2} + 9}$
Step-by-step solution
1. $L {f} = \int_{0}^{\infty} e^{- s x} f (x) d x$
2. $= \frac{s}{s ^{2} + 9}$
15.
Find the Laplace transform.
$L {e^{4 x}} (s)$
Answer: $\frac{1}{s - 4}$
Step-by-step solution
1. $L {f} = \int_{0}^{\infty} e^{- s x} f (x) d x$
2. $= \frac{1}{s - 4}$
16.
Find the Laplace transform.
$L {sin (4 x)} (s)$
Answer: $\frac{4}{s ^{2} + 16}$
Step-by-step solution
1. $L {f} = \int_{0}^{\infty} e^{- s x} f (x) d x$
2. $= \frac{4}{s ^{2} + 16}$
17.
Find the Laplace transform.
$L {cos (4 x)} (s)$
Answer: $\frac{s}{s ^{2} + 16}$
Step-by-step solution
1. $L {f} = \int_{0}^{\infty} e^{- s x} f (x) d x$
2. $= \frac{s}{s ^{2} + 16}$

The formulas you need here

Laplace transform of a derivative

17 worked problems

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