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Subsection 9.2.3 Power Function, tn

The power function tn is another common function type found in differential equations. The Laplace transform of tn follows a recursive pattern, which simplifies the computation for higher powers. We’ve already seen that L{1}=1/s. Now, let’s compute the transforms for t and t2.
If you look in each solution, before computing L=0, you’ll notice a relationship between the Laplace transforms of powers that differ by one. Namely,
L{t}=1sL{1}andL{t2}=2sL{t},
and if you compute the Laplace transform of t3, you’ll find that
L{t3}=3sL{t2}.
In general, this recursive pattern continues for any power n as
L{tn}=nsL{tn1}.
So if we wanted the Laplace transform of t4, we could find it like so
L{t4}= 4sL{t3}= 4s[3sL{t2}]=43s2[2sL{t}]=432s31s2=4321s5factorial.
This pattern is true for higher powers of t, leading to the next laplace transform rule which makes use of the factorial.

Common Laplace Transform (Power).

L3
L{tn}=n!sn+1,s>0,n=1,2,3,

Reading Questions Check-Point Questions

1. L{t4}=X!s5.

    L{t4}=X!s5
  • 4
  • Correct! The Laplace transform of t4 is 4!s5.
  • 5
  • No, try again.
  • 24
  • No, notice the factorial in the numerator.
  • 120
  • No, try again.

2. L{t3}=6X.

    L{t3}=6X
  • s4
  • No, the power of s in the denominator should be 4.
  • s4
  • Correct! The Laplace transform of t3 is 6s4.
  • s3
  • No, the power of s in the denominator should be 4.
  • s5
  • No, the power of s in the denominator should be 4.

3. L{X}= 479001600s13.

    L{X}= 479001600s13
  • t9
  • No, try again.
  • t10
  • No, try again.
  • t11
  • No, try again.
  • t12
  • Correct! The Laplace transform of t12 is 479001600s13.

4. L{t}= ?

    L{t}= ?
  • 1s2
  • Correct! The Laplace transform of t is 1s2.
  • 1s
  • No, the power of s in the denominator is not 1.
  • 1
  • No, the Laplace transform of t is not a constant.
  • 2s2
  • No, there should not be a 2 in the numerator.

5. L{t2}= ?

    L{t2}= ?
  • 1s2
  • Correct! The Laplace transform of t is 1s2.
  • 1s
  • No, the power of s in the denominator is not 1.
  • 1
  • No, the Laplace transform of t is not a constant.
  • 2s2
  • No, there should not be a 2 in the numerator.