Laplace Transform: Overview, Questions, Preparation

Differential Equations 2021 ( Differential Equations )

Rachit Kumar Saxena

Rachit Kumar SaxenaManager-Editorial

Updated on Jun 29, 2021 11:22 IST

What is Laplace transform?

Laplace Transform is the integral method of converting the given derivative function with real variable z into a complex function with variable p. You can solve the differential equations with constant coefficients with the Laplace transform operator. It is the best method to transform the Differential Equations into Algebraic Equations. The name of a one-sided transform also calls Laplace to transform. It is denoted by the function f(t) or F(p), and the equation is -

F(p) =   0 e.-p.z. f(z). dz

Here the z indicates time and z≥0 (always). 

Properties of Laplace Transform

Some well-known properties of Laplace transformation are:

If f1 (z) ⟷ F1 (p) and 

f2 (z) ⟷ F2 (p), 

[note: ⟷ means Laplace Transform]

Linearity

A f1(z) + B f2(z) ⟷ A F1(p) + B F2(p)

nth Derivative

(dn f(z)/ dzn) ⟷ pn F(p) − ni = 1 pn −i fi − 1 (0)

Frequency Shifting

ep0z f(z)) ⟷ F(p - p0)

Multiplication by Time

T f(z) ⟷ (−d F(p)⁄dp)

Time Reversal

f (-z) ⟷ F(-p)

Integration

z0 f(λ) dλ ⟷ 1⁄s F(p)

Complex Shift

f(z) e−at ⟷ F(p + a)

Time Scaling

f (z⁄a) ⟷ a F(ap)

Applications of Laplace Transform

  • It converts complex differential equations having polynomials into simpler equations.
  • Laplace also has its foot in various engineering tasks like System Modelling, Electrical Digital Signal Processing, Circuit Analysis, etc. 
  • Laplace transform also has its use in telecommunication for transferring the signals to both directions of the medium. 

Weightage of Laplace Transform

Laplace Transform is the topic of Differential Equations, and the chapter mostly arrives in class 12 exams. Nearly 3 to 4 questions arrive in the exam with the weightage of approximately seven marks. Other topics covered  in the Differential Equations are:

  • Differential equation order and degree
  • Homogeneous and Linear differential equations
  • Forming different equation that can represent a family of curves
  • Solving the first-degree differential equation

Apart from the Laplace Transform, we also have First and Second-Order Differential Equations. The equation explains the first-order differential equation: 

dy/dx =f (x, y) 

where x and y are two variables with their first derivative dy/dx, and f(x, y) is the function of the given variables. 

A Second-Order Differential Equation is written in the form of  d2y/dx2 = f(x, y). 

It is the differential of first-order derivative (d(dy/dx)).

Illustrated Examples of Laplace Transform

1. Let f(z) = eaz, z≥0

Solution.

Step1: L(eaz) =F(p) = 0 e.-p.z. f(z). dz= lim A→∞ A0 e-(p-a)z.dz

Step2: limA→∞ [-(e-(p-a)z.)/p-a] 0A  =  limA→∞{[1/(p-a)] - [e-(p-a)A/(p-a)]}

Step3: 1/p-a, for all p>a.

2. Let f(z) = 1, z≥0

Solution.

Step 1: L(1) = F(p) = 0 e-p.z. 1.dz = limA→∞A0 e -p.z.dz

Step 2: limA→∞ -e-pz/p  0A = limA→∞ [1/p - e-Ap/p] = 1/p, for all p>0

3. Find the laplace transform of  f(z) = z2.e-2x.cos 3z

Solution.

Step1: Let g(z) = cos 3z, h(z) = e-2x.cos 3z = e-2x.g(z), 
therefore f(z) = z2.h(z)

Step2: Let G(p) = L{g(z)}, H(p) = L{h(z)}, F(s) = L{f(t)}

Step3: Therefore, 

G(p) = p/(p2+9), H(p) = p+2/((s+2)2+9},

F(p) = -d/dp[-d(H(p))/dp] = {2(s+2)(s2+4s-23)}/(s2+4s+13)3

FAQs on Laplace Transform

Q: In which field do we use the Laplace Transform?

A: Laplace Transform has its use in various fields but mostly in the engineering field.

Q: How is the Laplace transform used in astronomy?

A: It is used in astronomy for determining the astronomical object's shape.

Q: What will be the Laplace transform of sin z?

A: f(z) = sin z is the assumed function which satisfies all the conditions and its Laplace Transform will be L{sin z} = 1/(p^2 + 1). As we are familiar with the Laplace transform of sin bz = b/(s^2 + b^2). 

Q: What are the uses of Laplace Transform in Daily Life?

A: Communicating Devices uses the Laplace Transform's application to convert the time-varying waves into frequency function.

Q: What is Inverse Laplace Transform?

A: It is converting a Laplace Transform F(p) into a time function f(z).

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