MAST20029 · Engineering Mathematics
Laplace Transforms
The Laplace transform turns differential equations into algebra: transform the IVP, solve for F(s), then invert with the provided table and partial fractions. This chapter covers transforms of derivatives, the s-shifting and t-shifting theorems, unit step and Dirac delta functions for piecewise and impulsive forcing, and the convolution theorem for integral equations. The exam (Q6) explicitly requires you to state which theorem (shift or convolution) you used.
What this chapter covers
- 011. Definition L{f} = ∫₀^∞ f(t) e^(−st) dt and the standard transform table
- 022. Transforms of derivatives: L{f'} = sF − f(0), L{f''} = s²F − sf(0) − f'(0)
- 033. Solving IVPs: transform → solve algebraically for F(s) → invert
- 044. s-shifting theorem: L{e^(−at) f(t)} = F(s + a)
- 055. Unit step u(t−a) and t-shifting: L{f(t−a) u(t−a)} = e^(−as) F(s)
- 066. Dirac delta δ(t−a): L{δ(t−a)} = e^(−as) for impulsive forcing
- 077. Convolution theorem: L{(f∗g)(t)} = F(s) G(s) for integral equations
- 088. Stating which theorem (shift / convolution) you used — exam requirement
IVP with a Dirac delta forcing via Laplace
- 1 markTransform, using L{x''} = s²X − s x(0) − x'(0), L{x'} = sX − x(0), and L{δ(t−2)} = e^(−2s): (s²X − s) + 3(sX − 1) = e^(−2s).
- 1 markSolve for X: X(s² + 3s) = s + 3 + e^(−2s), so X = (s + 3)/(s(s + 3)) + e^(−2s)/(s(s + 3)) = 1/s + e^(−2s)/(s(s + 3)).
- 1 markPartial fractions on the second term: 1/(s(s + 3)) = (1/3)(1/s − 1/(s + 3)).
- 1 markInvert. The 1/s term gives 1. For the e^(−2s) factor use the t-shifting theorem (state it): e^(−2s)/(s(s+3)) inverts to (1/3)(1 − e^(−3(t−2))) u(t − 2).
- 2 marksAssemble: x(t) = 1 + (1/3)(1 − e^(−3(t−2))) u(t − 2), and state that the t-shifting theorem was used.
Key terms
- Laplace transform
- L{f} = F(s) = ∫₀^∞ f(t) e^(−st) dt, a map from a function of t to a function of s that converts differentiation into multiplication by s.
- Transform of a derivative
- L{f'} = sF(s) − f(0) and L{f''} = s²F(s) − sf(0) − f'(0). These build the initial conditions directly into the algebra when solving an IVP.
- s-shifting theorem
- L{e^(−at) f(t)} = F(s + a): multiplying by e^(−at) in t shifts the transform variable by a. Used to invert terms like 1/((s+a)²+b²).
- t-shifting theorem
- L{f(t−a) u(t−a)} = e^(−as) F(s): a delay by a in t corresponds to an e^(−as) factor in s. The presence of e^(−as) signals a shifted step in the inverse.
- Dirac delta δ(t−a)
- An idealised unit impulse at t = a, with L{δ(t−a)} = e^(−as). It models a sudden kick (a hammer blow) in the forcing of an ODE.
- Convolution theorem
- L{(f∗g)(t)} = F(s) G(s), where (f∗g)(t) = ∫₀ᵗ f(τ) g(t−τ) dτ. It converts an integral equation into algebra in the transform domain.
Laplace Transforms FAQ
How does the Laplace method solve an ODE?
Transform both sides of the IVP, which turns derivatives into algebra and folds in the initial conditions via L{f'} = sF − f(0) and L{f''} = s²F − sf(0) − f'(0). Solve the resulting algebraic equation for F(s), then invert using the provided table, partial fractions, and the shift theorems to recover f(t).
When do I use the t-shifting theorem versus s-shifting?
Use t-shifting (L{f(t−a)u(t−a)} = e^(−as)F(s)) when you see an e^(−as) factor or a delayed/piecewise forcing — the inverse will contain a unit step u(t−a). Use s-shifting (L{e^(−at)f(t)} = F(s+a)) when the transform has s replaced by s+a, typically from an e^(−at) factor in t. The exam asks you to name which one you used.
How do I handle a Dirac delta in the forcing?
Transform δ(t−a) to e^(−as), solve for X(s) as usual, and invert. The e^(−as) factor means you apply the t-shifting theorem on inversion, producing a u(t−a) step in the answer. Keep the initial-condition terms from x'' and x' through the algebra.
What is the convolution theorem for?
It solves integral equations where the unknown y(t) appears under an integral ∫₀ᵗ y(τ) g(t−τ) dτ — that integral is a convolution, whose transform is the product Y(s)G(s). Transforming turns the whole equation into algebra you solve for Y(s), then invert. Always state that you used the convolution theorem.
Exam move
Make the Laplace pipeline automatic: transform (carrying initial conditions through f' and f''), solve for F(s), then invert with the table, partial fractions and the shift theorems. Learn to read the signals — an e^(−as) factor or piecewise forcing means t-shifting and a unit step in the answer; a convolution integral means the convolution theorem and a product of transforms. Practise partial fractions quickly, since most inversions hinge on them. Because the exam explicitly requires it, write down which theorem (s-shift, t-shift or convolution) you used at each step — those statements are marks.
