Differential Equations Laplace Transforms and Operational Methods Syllabus

Laplace Transforms and Operational Methods syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Laplace Transforms and Operational Methods in Differential Equations, not a summary of it.

1. 1

   The Laplace Transform

   3 topics

   • Definition and Existence
   • Transforms of Elementary Functions
   • Linearity and Basic Properties
2. 2

   Inverse Transforms and Operational Properties

   4 topics

   • Inverse Laplace Transform
   • Transforms of Derivatives and Integrals
   • Translation Theorems
   • Derivatives of Transforms
3. 3

   Solving Initial-Value Problems

   2 topics

   • Laplace Method for IVPs
   • Discontinuous and Periodic Forcing
4. 4

   Special Functions and Convolution

   3 topics

   • Dirac Delta Function
   • Convolution Theorem
   • Transform of a Convolution

Laplace Transforms and Operational Methods flashcards for Differential Equations

21 of 50 cards from the Laplace Transforms and Operational Methods deck — real questions with worked answers.

1. What is the definition of the Laplace transform of a function $f(t)$ defined for $t \geq 0$?

   $$\mathcal{L}\{f(t)\} = F(s) = \int_{0}^{\infty} e^{-st} f(t)\, dt$$ for all values of $s$ for which the improper integral converges.

2. What two conditions guarantee the existence of the Laplace transform of $f(t)$ (Existence Theorem)?

   $f$ must be (1) piecewise continuous on every finite interval $[0, N]$, and (2) of exponential order, i.e. there exist constants $M > 0$, $c$, and $T$ such that $|f(t)| \leq M e^{ct}$ for all $t \geq T$. Then $F(s)$ exists for $s > c$.

3. What does it mean for a function $f(t)$ to be of exponential order $c$?

   There exist constants $M > 0$ and $T > 0$ such that $|f(t)| \leq M e^{ct}$ for all $t \geq T$; the function grows no faster than an exponential.

4. Give an example of a function that does NOT have a Laplace transform because it is not of exponential order.

   $f(t) = e^{t^{2}}$ grows faster than any $M e^{ct}$, so it is not of exponential order and its Laplace transform does not exist.

5. What is the limiting behavior of $F(s)$ as $s \to \infty$ for any function of exponential order?

   $$\lim_{s \to \infty} F(s) = 0.$$ Consequently, a function like $F(s) = \frac{s}{s+1}$ (which tends to $1$) cannot be the Laplace transform of a piecewise-continuous function of exponential order.

6. What is the Laplace transform of the constant function $f(t) = 1$, and for what $s$?

   $$\mathcal{L}\{1\} = \frac{1}{s}, \quad s > 0.$$

7. What is the Laplace transform of $f(t) = t^{n}$ for a positive integer $n$?

   $$\mathcal{L}\{t^{n}\} = \frac{n!}{s^{n+1}}, \quad s > 0.$$

8. What is the Laplace transform of $t^{a}$ for real $a > -1$, using the Gamma function?

   $$\mathcal{L}\{t^{a}\} = \frac{\Gamma(a+1)}{s^{a+1}}, \quad s > 0,$$ where $\Gamma(a+1) = \int_{0}^{\infty} x^{a} e^{-x}\, dx$.

9. What is the Laplace transform of $f(t) = e^{at}$?

   $$\mathcal{L}\{e^{at}\} = \frac{1}{s-a}, \quad s > a.$$

10. What are the Laplace transforms of $\sin(kt)$ and $\cos(kt)$?

    $$\mathcal{L}\{\sin kt\} = \frac{k}{s^{2}+k^{2}}, \qquad \mathcal{L}\{\cos kt\} = \frac{s}{s^{2}+k^{2}}, \quad s > 0.$$

11. What are the Laplace transforms of $\sinh(kt)$ and $\cosh(kt)$?

    $$\mathcal{L}\{\sinh kt\} = \frac{k}{s^{2}-k^{2}}, \qquad \mathcal{L}\{\cosh kt\} = \frac{s}{s^{2}-k^{2}}, \quad s > |k|.$$

12. State the linearity property of the Laplace transform.

    For constants $\alpha, \beta$: $$\mathcal{L}\{\alpha f(t) + \beta g(t)\} = \alpha \mathcal{L}\{f(t)\} + \beta \mathcal{L}\{g(t)\}.$$ The Laplace transform is a linear operator.

13. Is the inverse Laplace transform a linear operator?

    Yes. $$\mathcal{L}^{-1}\{\alpha F(s) + \beta G(s)\} = \alpha \mathcal{L}^{-1}\{F(s)\} + \beta \mathcal{L}^{-1}\{G(s)\}.$$

14. What is the inverse Laplace transform, and how is it defined notationally?

    If $\mathcal{L}\{f(t)\} = F(s)$, then $f(t) = \mathcal{L}^{-1}\{F(s)\}$ is the inverse Laplace transform. It recovers the time-domain function whose transform is $F(s)$.

15. What does Lerch's theorem say about the uniqueness of inverse Laplace transforms?

    If two continuous functions of exponential order have the same Laplace transform, they are identical. Thus the inverse transform is essentially unique among continuous functions (it may differ only at isolated discontinuity points of piecewise-continuous functions).

16. What is $\mathcal{L}^{-1}\left\{\frac{1}{s^{n+1}}\right\}$?

    $$\mathcal{L}^{-1}\left\{\frac{1}{s^{n+1}}\right\} = \frac{t^{n}}{n!}.$$

17. What algebraic technique is most commonly used to compute an inverse Laplace transform of a rational function $\frac{P(s)}{Q(s)}$?

    Partial fraction decomposition: split the rational function into simpler terms whose inverse transforms are known from a table, then apply linearity.

18. State the Laplace transform of the first derivative $f'(t)$.

    $$\mathcal{L}\{f'(t)\} = sF(s) - f(0).$$

19. State the Laplace transform of the second derivative $f''(t)$.

    $$\mathcal{L}\{f''(t)\} = s^{2}F(s) - s f(0) - f'(0).$$

20. State the general formula for the Laplace transform of the $n$-th derivative $f^{(n)}(t)$.

    $$\mathcal{L}\{f^{(n)}(t)\} = s^{n}F(s) - s^{n-1}f(0) - s^{n-2}f'(0) - \cdots - f^{(n-1)}(0).$$

21. What is the Laplace transform of an integral, $\mathcal{L}\left\{\int_{0}^{t} f(\tau)\, d\tau\right\}$?

    $$\mathcal{L}\left\{\int_{0}^{t} f(\tau)\, d\tau\right\} = \frac{F(s)}{s}.$$

See more Laplace Transforms and Operational Methods flashcards →

Planning Laplace Transforms and Operational Methods for Differential Equations

Laplace Transforms and Operational Methods (Differential Equations) FAQ