Linearity Property of Laplace Transform
The linearity property of Laplace transforms states that the Laplace transform of the weighted sum of two signals is equal to the weighted sum of the individual sum Laplace transforms.
Consider the functions x_{1}(t) and x_{2}(t) whose Laplace transform exists.
If a and b are constants, then, according to the linearity property of Laplace transform,
Proof
Now let's learn how we can prove the linearity property of Laplace transform.
According to the definition of Laplace transform,
This is the same as,
Numerical Examples
Let's discuss some problems based on the linearity property of Laplace transform.
1. Find the Laplace transform of f(t) = 10t−5.
Solution:
L[10t − 5] = 10L(t) − 5L(1) [Using linearity property of Laplace transform]
The Laplace transform of the functions are as follows:
10L(t) = 10(1/s^{2})
5L(1) = 5(1/s)
∴ L[10t − 5] = 10(1/s^{2}) − 5(1/s)
L[10t − 5] = 10/s^{2} − 5/s
L[10t − 5] = (10  5s)/s^{2}
2. Find the Laplace transform of x(t) = 2e^{−5t}u(t) − 15e^{4t}u(−t).
Solution:
L[2e^{−5t}u(t) − 15e^{4t}u(−t)] = 2L[e^{−5t}u(t)] − 15L[e^{4t}u(−t)] [Using linearity property of Laplace transform]
The Laplace transform of the functions are as follows:
2L[e^{−5t}u(t)] = 2/(s + 5)
15L[e^{4t}u(−t)] = 15/(s  4)
∴ L[2e^{−5t}u(t) − 15e^{4t}u(−t)] = 2/(s + 5) − 15/(s  4)
L[2e^{−5t}u(t) − 15e^{4t}u(−t)] = (17s  83)/(s^{2}+s−20)
3. By using the linearity property of Laplace transform, show that
L(cosh at) = s/(s^{2}−a^{2})
Solution:
cosh at = (e^{at }+ e^{−at)}/2
L(cosh at) = L(e^{at})/2 + L(e^{at})/2
The Laplace transform of the functions are as follows:
L(e^{at}) = 1/(s  a)
L(e^{at}) = 1/(s + a)
∴ L(cosh at) = L(e^{at})/2 + L(e^{at})/2
L(cosh at) = (1/(s  a))/2 + (1/(s + a))/2
L(cosh at) = (1/(s  a) + 1/(s + a))/2
L(cosh at) = s/(s^{2}−a^{2})
Hence Proved.
Frequently Asked Questions

Is Laplace transform a linear operator?
Yes, the Laplace transform is a linear operator with the linearity property.

Can the linearity property of Laplace transform be extended?
Yes, the linearity property of Laplace transform can be extended to more than two functions.

Name some properties of Laplace transform.
Some properties of Laplace transform are frequency shifting property, multiplication by time, time scaling property, time reversal property, etc.

How can we prove the linearity property of Laplace transform?
The linearity property of Laplace transform can be proved by using the definition of Laplace transform.
Conclusion
In this article, we have extensively discussed the linearity property of Laplace transform along with some numerical examples.
Check out this problem  Subarray Sum Divisible By K
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