Subsection 9.2.3 Power Function,
The power function is another common function type found in differential equations. The Laplace transform of follows a recursive pattern, which simplifies the computation for higher powers. We’ve already seen that Now, let’s compute the transforms for and
If you look in each solution, before computing you’ll notice a relationship between the Laplace transforms of powers that differ by one. Namely,
and if you compute the Laplace transform of you’ll find that
In general, this recursive pattern continues for any power as
So if we wanted the Laplace transform of we could find it like so
Common Laplace Transform (Power).
Reading Questions Check-Point Questions
1. .
- Correct! The Laplace transform of
is - No, try again.
- No, notice the factorial in the numerator.
- No, try again.
2. .
- No, the power of
in the denominator should be - Correct! The Laplace transform of
is - No, the power of
in the denominator should be - No, the power of
in the denominator should be
3. .
- No, try again.
- No, try again.
- No, try again.
- Correct! The Laplace transform of
is
4. ?
- Correct! The Laplace transform of
is - No, the power of
in the denominator is not - No, the Laplace transform of
is not a constant. - No, there should not be a
in the numerator.
5. ?
- Correct! The Laplace transform of
is - No, the power of
in the denominator is not - No, the Laplace transform of
is not a constant. - No, there should not be a
in the numerator.