Rachit Kumar SaxenaManager-Editorial
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) − n∑i = 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 |
z∫0 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→∞ A∫0 e-(p-a)z.dz
Step2: limA→∞ [-(e-(p-a)z.)/p-a] 0│A = 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→∞A∫0 e -p.z.dz
Step 2: limA→∞ -e-pz/p 0│A = 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?
Q: How is the Laplace transform used in astronomy?
Q: What will be the Laplace transform of sin z?
Q: What are the uses of Laplace Transform in Daily Life?
Q: What is Inverse Laplace Transform?
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