Question

Find the Laplace transform of each of the following expressions: (a) \(t-3\) (b) \(2 t^{3}+5 t\) (c) \(7-3 t^{4}\) (d) \(\sin 2 t+2 \sin t\) (e) \(\cos t+t\)

Short answer

Question: Find the Laplace transform of the following functions: (a) \(t-3\), (b) \(2t^3 + 5t\), (c) \(7 - 3t^4\), (d) \(\sin 2t + 2\sin t\), and (e) \(\cos t + t\). Solution: (a) The Laplace transform of \(t-3\) is \(\frac{1}{s^2}+\frac{3}{s}\) (b) The Laplace transform of \(2t^3 + 5t\) is \(\frac{12}{s^4}+\frac{5}{s^2}\) (c) The Laplace transform of \(7-3t^4\) is \(\frac{7}{s}-\frac{72}{s^5}\) (d) The Laplace transform of \(\sin 2t + 2\sin t\) is \(\frac{2}{s^2+4}+\frac{2}{s^2+1}\) (e) The Laplace transform of \(\cos t + t\) is \(\frac{s}{s^2+1}+\frac{1}{s^2}\)

Step-by-step solution

Identify the Laplace transform formula for individual functions

In this case, we have a linear combination of two functions \(t\) and \(-3\). The Laplace transform is a linear operator, which means that \(\mathcal{L}\{t-3\} = \mathcal{L}\{t\} - \mathcal{L}\{-3\}\). To find the Laplace transform of \(t-3\), we will need to compute the Laplace transform of each function individually. The formula for the Laplace transform of polynomials is given by: $$\mathcal{L}\{t^n\}=\frac{n!}{s^{n+1}}$$ For the constant function, the Laplace transform is given by: $$\mathcal{L}\{c\}=\frac{c}{s}$$ Where c is a constant.

Compute the Laplace transform of each function

Applying the formulas above, we can compute the Laplace transform for t and -3: $$\mathcal{L}\{t\}=\frac{1!}{s^{1+1}}=\frac{1}{s^2}$$ $$\mathcal{L}\{-3\} = \frac{-3}{s}$$

Combine the Laplace transforms

Now, we can combine the results to find the Laplace transform of \(t-3\): $$\mathcal{L}\{t-3\} = \mathcal{L}\{t\} - \mathcal{L}\{-3\} = \frac{1}{s^2} - \frac{-3}{s} = \frac{1}{s^2} + \frac{3}{s}$$ (b) Laplace transform of \(2t^3 + 5t\):

Identify the Laplace transform formula for individual functions

In this case, we have the functions \(2t^3\) and \(5t\). Since the Laplace transform is a linear operator, we can compute the Laplace transform of each individual function and then combine them: $$\mathcal{L}\{2t^3 + 5t\} = 2\mathcal{L}\{t^3\} + 5\mathcal{L}\{t\}$$

Compute the Laplace transform of each function

Using the Laplace transform formula for polynomials, we find the Laplace transforms for \(t^3\) and \(t\): $$\mathcal{L}\{t^3\}=\frac{3!}{s^{3+1}}=\frac{6}{s^4}$$ $$\mathcal{L}\{t\}=\frac{1!}{s^{1+1}}=\frac{1}{s^2}$$

Combine the Laplace transforms

Now, we can combine the results with the constants to find the Laplace transform of \(2t^3 + 5t\): $$\mathcal{L}\{2t^3 + 5t\} = 2\cdot\frac{6}{s^4} + 5\cdot\frac{1}{s^2} = \frac{12}{s^4} + \frac{5}{s^2}$$ (c) Laplace transform of \(7 - 3t^4\): This problem is similar to parts (a) and (b), so we can directly write the Laplace transform using the polynomial formula: $$\mathcal{L}\{7 -3t^4\} = 7\mathcal{L}\{1\} - 3\mathcal{L}\{t^4\} = 7\cdot\frac{1}{s} - 3\cdot\frac{4!}{s^5} = \frac{7}{s} - \frac{72}{s^5}$$ (d) Laplace transform of \(\sin 2t + 2\sin t\):

Identify the Laplace transform formula for individual functions

The Laplace transform for sinusoidal functions is given by: $$\mathcal{L}\{\sin{at}\}=\frac{a}{s^2+a^2}$$ Using linearity, we can compute the Laplace transform of the given expression as: $$\mathcal{L}\{\sin 2t + 2\sin t\} = \mathcal{L}\{\sin 2t\} + 2\mathcal{L}\{\sin t\}$$

Compute the Laplace transform of each function

Substituting the appropriate values, we have: $$\mathcal{L}\{\sin 2t\} = \frac{2}{s^2+2^2} = \frac{2}{s^2+4}$$ $$\mathcal{L}\{\sin t\} = \frac{1}{s^2+1^2} = \frac{1}{s^2+1}$$

Combine the Laplace transforms

Now, combine the results: $$\mathcal{L}\{\sin 2t + 2\sin t\} = \frac{2}{s^2+4} + 2\cdot\frac{1}{s^2+1} = \frac{2}{s^2+4} + \frac{2}{s^2+1}$$ (e) Laplace transform of \(\cos t + t\):

Identify the Laplace transform formula for individual functions

The Laplace transform for cosine functions is given by: $$\mathcal{L}\{\cos{at}\}=\frac{s}{s^2+a^2}$$ Using linearity, we can compute the Laplace transform of the given expression as: $$\mathcal{L}\{\cos t + t\} = \mathcal{L}\{\cos t\} + \mathcal{L}\{t\}$$

Compute the Laplace transform of each function

Substituting the appropriate values, we have: $$\mathcal{L}\{\cos t\} = \frac{s}{s^2+1^2} = \frac{s}{s^2+1}$$ $$\mathcal{L}\{t\} = \frac{1!}{s^{1+1}} = \frac{1}{s^2}$$

Combine the Laplace transforms

Now, combine the results: $$\mathcal{L}\{\cos t + t\} = \frac{s}{s^2+1} + \frac{1}{s^2}$$

Key concepts

Linear Operator

In mathematics, a linear operator is a special type of function between two vector spaces that preserves the operations of vector addition and scalar multiplication. When dealing with Laplace transforms, it is important to understand that they are linear operators. This means they possess properties that make them easier to work with when functions are combined, like addition or multiplication by a constant.

Imagine you're adding two functions, say \(f(t)\) and \(g(t)\). If you want to find the Laplace transform of this combined function, you simply take the Laplace transform of each function individually and add the results. Mathematically, this looks like:

• \(\mathcal{L}\{af(t) + bg(t)\} = a\mathcal{L}\{f(t)\} + b\mathcal{L}\{g(t)\}\)

Here, \(a\) and \(b\) are constants and \(\mathcal{L}\) stands for the Laplace transform. This linearity property is extremely useful because it allows us to break down complex expressions into simpler parts, making computations much less daunting.

Polynomial Functions

Polynomial functions are some of the most common functions you'll encounter in calculus. These functions are expressions consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. They look like this: \(a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\).

• Each term follows the form \(a_i x^i\), where \(a_i\) is a constant and \(i\) is a non-negative integer exponent.

Now, when you're applying the Laplace transform to polynomial functions, there's a neat formula you can use:

• \(\mathcal{L}\{t^n\} = \frac{n!}{s^{n+1}}\)

This formula provides a quick way to handle each term of the polynomial. You simply take the factorial of the exponent, then divide by \(s\) raised to one more than the exponent. Understanding these fundamentals makes polynomial transformations straightforward.

Sinusoidal Functions

Sinusoidal functions include sine and cosine functions, which are pivotal in the study of periodic phenomena. They play a critical role in describing oscillations and waves. In Laplace transforms, these functions are also handled through specific formulas. For sine functions, the Laplace transform formula is:

• \(\mathcal{L}\{\sin(at)\} = \frac{a}{s^2 + a^2}\)

This formula simplifies calculations by producing a function of \(s\) based on the frequency parameter \(a\). This form allows the frequency component to influence how the sine wave is transformed.

When you have expressions involving multiple sine functions, like \(\sin(2t) + 2\sin(t)\), you can leverage the linear operator property. Split and transform them one at a time before summing them up. This leads to structured and efficient computations, especially in complex systems.

Cosine Functions

Cosine functions, closely related to sine functions, are also essential in analyzing oscillatory behavior. The Laplace transform handles cosine functions through a special formula just like sine. The basic formula for the Laplace transform of a cosine function is:

• \(\mathcal{L}\{\cos(at)\} = \frac{s}{s^2 + a^2}\)

This equation, similar to that for the sine function, transforms the time domain cosine function into a function of \(s\), reflecting the effect of the frequency \(a\).

Harmonically, sine and cosine functions form the backbone of Fourier series, but within the Laplace domain, they're treated with respect to their frequency components. As an illustration, if you need to transform expressions like \(\cos(t) + t\), you evaluate the Laplace transform of each function individually and sum them up. Using such techniques allows you to interpolate complex oscillatory functions seamlessly.