Laplace transform · Definition

Laplace transform of a derivative

differential equationsLaplace transform

Before you start you need to know

This is the rule that makes the Laplace transform useful for differential equations: a derivative becomes multiplication by \(s\).

Definition

The Laplace transform \(\mathcal L\) of the derivative \(f'\) of a function \(f(t)\) with transform \(F(s)\) is an expression in \(F(s)\) and the initial value \(f(0)\).

\[\mathcal L\{f'\}(s)=sF(s)-f(0)\]

The rule comes from integration by parts: \(\int_0^\infty e^{-st}f'(t)\,dt=\big[e^{-st}f(t)\big]_0^\infty+s\int_0^\infty e^{-st}f(t)\,dt\). Since \(f\) is of exponential order, the term at infinity vanishes for large \(s\) and only the term at \(t=0\) remains, with a negative sign, hence \(-f(0)\).

Applying the rule twice gives \(\mathcal L\{f''\}=s^2F(s)-sf(0)-f'(0)\), and in general \(\mathcal L\{f^{(n)}\}=s^nF(s)-s^{n-1}f(0)-\cdots-f^{(n-1)}(0)\). Each derivative adds a factor \(s\) and uses up one initial condition; that is why the method asks for exactly the data of an initial value problem.

Guiding example: in \(y'+2y=0\), \(y(0)=3\), transforming term by term gives \(sY-3+2Y=0\). The differential equation became an algebraic equation in \(Y(s)\): \(Y=3/(s+2)\). What remains is going back to \(y(t)\) with the inverse transform, the next step of the method.

A typical mistake is forgetting the initial value or giving it the wrong sign. Another is believing that \(\mathcal L\{f'\}\) is simply \(sF(s)\): that holds only when \(f(0)=0\), as happens with \(f(t)=t^2\).

Think about it

Why is no general solution needed before applying initial conditions?

Because \(f(0)\) and \(f'(0)\) appear directly in the transform of the derivatives: the initial data enter from the first step and the solution obtained already satisfies them.

If \(f(0)=0\), what does differentiating \(f\) do on the \(s\) side?

Differentiating amounts to multiplying the transform by \(s\): \(\mathcal L\{f'\}=sF(s)\).

Exercises 10

  1. If \(f(t)=e^{3t}\), then \(F(s)=\dfrac1{s-3}\) and \(f(0)=1\). What is \(\mathcal L\{f'\}\)?
  2. Which is the correct formula for \(\mathcal L\{f''\}\)?
  3. For \(f(t)=\sin t\) we have \(F(s)=\dfrac1{s^2+1}\) and \(f(0)=0\). What is \(\mathcal L\{f'\}\)?
  4. Transforming \(y'+2y=0\), \(y(0)=3\), what is \(Y(s)=\mathcal L\{y\}\)?
  5. Transforming \(y'+y=1\), \(y(0)=0\), what \(Y(s)\) results?
  6. Transforming \(y''+4y=0\), \(y(0)=1\), \(y'(0)=0\), what \(Y(s)\) results?
  7. Let \(f(t)=t^2\). Using the derivative rule with \(F(s)=2/s^3\), what is \(\mathcal L\{f'\}\)?
  8. Transforming \(y'+3y=e^{-t}\), \(y(0)=1\), what \(Y(s)\) results?
  9. Transforming \(y''-3y'+2y=0\), \(y(0)=1\), \(y'(0)=0\), what \(Y(s)\) results?
  10. It is known that \(\mathcal L\{f''\}=s^2F(s)-2s+3\). What are \(f(0)\) and \(f'(0)\)?

References

  • William E. Boyce, Richard C. DiPrima, Douglas B. Meade, Elementary Differential Equations and Boundary Value Problems, 12th edition, Wiley, 2021. §6.2
  • Dennis G. Zill, A First Course in Differential Equations with Modeling Applications, 11th edition, Cengage, 2018. §7.2