Advanced Math Solutions – Laplace Calculator, Laplace Transform

In previous posts, we talked about the four types of ODE - linear first order, separable, Bernoulli, and exact. In today’s post, we will learn about Laplace Transforms, how to compute Laplace transforms and inverse Laplace transforms. I’ll admit I was afraid of Laplace transforms before I learned it. The new symbols and the messiness of the problem really intimidated me. However, once I learned what Laplace transforms are and how to do these types of problems, I began to realize that it isn’t so scary. With the assistance of a table and some formulas, anyone can do Laplace transforms.

What is a Laplace Transform?

Laplace transforms can be used to solve differential equations. They turn differential equations into algebraic problems.

Definition:
Suppose f(t) is a piecewise continuous function, a function made up of a finite number of continuous pieces. The Laplace transform of f(t) is denoted L{f(t)} and defined as:

L\left\{f(t)\right\}=\int_{0}^{\infty}e^{-st}f(t)dt

If you see L\left\{f(t)\right\}=\int_{-\infty}^{\infty}e^{-st}f(t)dt, then you can assume that for t\lt0,\quad f(t)=0, and then you can use the original definition of Laplace transform.


Now, we will get into how to compute Laplace transforms:

Laplace transforms can be computed using a table and the linearity property, “Given f(t) and g(t) then, L\left\{af(t)+bg(t)\right\}=aF(s)+bG(s).” The statement means that after you’ve taken the transform of the individual functions, then you can add back any constants and add or subtract the results.

Look at the table and see what functions you can transform. Algebraic manipulation may be required. The table will be your savior when it comes to these problems.

Simple enough! Let’s look at an example (click here):

This problem is very simple. It requires looking at the different functions, finding the corresponding transforms in the table, and then adding any constants and adding and subtracting the results together.


Here’s another example (click here):

Let’s get into inverse Laplace transforms!

What is an inverse Laplace transform?

An inverse Laplace transform is when we are given a transform, F(s), and asked what function(s) we had originally.
Definition:

f(t)=L^{-1}\left\{F(s)\right\}

How to compute the inverse Laplace transforms:

Just like Laplace transforms have a linearity property, so do inverse Laplace transforms. “Given the two Laplace transforms F(s) and G(s), then L^{-1}\left\{aF(s)+bG(s)\right}=aL^{-1}\left\{F(s)\right\}+bL^{-1}\left\{G(s)\right\}.”

When trying to computer the inverse Laplace transforms, it is important to first look at the denominator and then try to identify the transform based on that. If you can’t figure it out just based on looking at the denominator, look at the numerator. Sometime you may have to manipulate the numerator to get into the correct form needed.

Let’s see an example (click here):

By looking at the table, we can see that the denominator is almost the same as the denominator of the transform for \sqrt{t}. With some algebraic manipulation to the numerator, we are able to figure out the inverse Laplace transform.

Here’s another example (click here):

See, Laplace transforms aren’t that hard after all. They can get a little messy and can be take a long time to solve after first, but with more practice, the better you’ll get. Make sure you have that table handy!

Until next time,
Leah

Advanced Math Solutions – Ordinary Differential Equations Calculator, Exact Differential Equations

In the previous posts, we have covered three types of ordinary differential equations, (ODE). We have now reached the last type of ODE. In this post, we will talk about exact differential equations.

What is an exact differential equation?

There must be a 0 on the right side of the equation and M(x,y)dx and N(x,y)dx must be separated by a +.

Steps to solve exact differential equations:

1. Verify that \frac{∂M(x,y)}{∂y}=\frac{∂N(x,y)}{∂x}
• Find M(x,y) and N(x,y)
2. Integrate \int M(x,y)dx or \int N(x,y)dy
• This will help us find Ψ(x,y)
3. Replace c with ƞ(x) if you integrated N(x,y) with respect to y, or ƞ(y) if you integrated M(x,y) with respect to x
4. Compute ƞ(x) or ƞ(y)
5. Solve to get the implicit or explicit solution, depending on which is preferred
• Don’t forget to substitute ƞ(x) or ƞ(y)

Exact differential equations can be tricky. We will solve the first example step by step to help you better understand how to solve exact differential equations.


First example (click here):

1. Verify that \frac{∂M(x,y)}{∂y}=\frac{∂N(x,y)}{∂x}

M(x,y)=2xy-9x^2

N(x,y)=2y+x^2+1

2. Integrate \int N(x,y)dy

3. Replace c with ƞ(x)

4. Compute ƞ(x)

We took the derivative with respect to x of Ψ(x,y), which is equal to y+x^2 y+y^2+ƞ(x). We then compared the derivative to M(x,y), the equation we didn’t integrate. We then integrated both sides to solve for ƞ(x).

5. Find the implicit equation

Here’s another example (click here):

Exact differential equations may look scary because of the odd looking symbols and multiple steps. If you double check your work, memorize the steps, and practice, you can definitely get this concept down. Don’t be afraid and dive in!

Until next time,

Leah