The Laplace Transform

Turning calculus into algebra, then turning it back.

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

What this lesson teaches

The student states what the Laplace transform does and uses a small table of transforms.

The Laplace transform sends a function of t to a function of s, turning derivatives into multiplication.
Transform the equation, solve algebraically for Y, then invert to recover y.
Initial conditions enter during the transform, so no constants remain to find at the end.

Straightforward practice. Get the method working first.

5 problems

Y = 4/(s + 3). The solution is Ce^(kt). What is k?
Answer -3
Why −3.
What does the Laplace transform do to differentiation?
Answer It turns it into multiplication by s, so calculus becomes algebra.
Why Differentiation becomes multiplication.
The transform of the constant 1 is 1 over s to what power?
Answer 1
Why Just 1/s.
The transform of t is 1 over s to what power?
Answer 2
Why 1/s².
The transform of e^(6t) is 1/(s − a). What is a?
Answer 6
Why The exponent.

The same ideas with more to keep track of.

3 problems

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

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

4 problems

Order the steps of solving with the Laplace transform.
Answer 1. Transform the whole equation into the s domain 2. Substitute the initial conditions as they appear 3. Solve the algebraic equation for Y 4. Take the inverse transform to recover y
Why The inverse comes last.
Under the Laplace transform, differentiation becomes which operation? Enter 1 multiplication, 2 addition.
Answer 1
Why Multiplying by s.
The Table: The transform of e^(4t) is 1/(s − a). What is a?
Answer 4
Why 4.
The Inverse: Y = 7/(s + 2). The solution is Ce^(kt). What is C?
Answer 7
Why 7.