The Laplace Transform

Turning calculus into algebra, then turning it back.

12 problems · about 23 minutes · F-IF.B.6

NameDate

Figure — use these to answer the problems

The transform of e^(at) is 1/(s − a). Change the rate a and the pole of the transform moves with it.

y = 1 · 0.5^x + 0

the coefficient1

a, the rate0.5

the time shown1

Figure 1

Straightforward practice. Get the method working first.

5 problems

Y = 4/(s + 3). The solution is Ce^(kt). What is k?
Answer
2
What does the Laplace transform do to differentiation?
- It turns it into multiplication by s, so calculus becomes algebra.
- It removes it entirely with no trace.
The transform of the constant 1 is 1 over s to what power?
Answer
The transform of t is 1 over s to what power?
Answer
The transform of e^(6t) is 1/(s − a). What is a?
Answer

The same ideas with more to keep track of.

3 problems

Y = 3/(s + 5). The solution is Ce^(kt). What is k?
Answer
The transform of y′ is sY minus what quantity? Enter 1 for the initial value, 2 for zero.
Answer
Y = 9/(s − 1). The solution is Ce^(kt). What is C?
Answer

Mixed problems. Work out what kind of question it is before you start.

4 problems

9
Order the steps of solving with the Laplace transform.
Write 1 to 4 in the boxes to put these in order.
- Transform the whole equation into the s domain
- Substitute the initial conditions as they appear
- Solve the algebraic equation for Y
- Take the inverse transform to recover y
Under the Laplace transform, differentiation becomes which operation? Enter 1 multiplication, 2 addition.
Answer
The Table
The transform of e^(4t) is 1/(s − a). What is a?
Answer
The Inverse
Y = 7/(s + 2). The solution is Ce^(kt). What is C?
Answer