The differential equation dy/dx=e^y(3x^2-6x) has a familiar look to it.
In fact, this equation was the first one to do so! In the 1700s, people used this type of equation to solve heat transfer problems. Today, it is still used in this way.
The problem is, it is very hard to find solutions to. Only one in a billion people know how to solve it!
The reason why this equation looks familiar is because it has a few variables that are changed. The unknown variable is x.
Solving differential equations
Many students begin analyzing differential equations as a way to learn how to solve them. However, there are much smarter ways to analyze equations.
Heuristic Hint: Don’t be afraid to make up solutions as you go. Even the best answers are not always the right ones at first!
There are many ways to analyze an equation. This is due in part to the fact that an equation can be a place where something is equal to another thing, or something can be different from another thing.
Solving differential equations using techniques such as calculus or linear algebra will not help when trying to solve a problem on your own. Many times, looking for new ways to solve an equation will help prevent yourself from relying on only one method of solving it.
blade-style: pull out a solution that you like and stick with it.
The general form of a differential equation
There are three types of differential equations: linear, exponential, and nonlinear. Each one presents a different challenge to a theoretical physicist.
Linear equations have a known solution (a line or a curve) and are easy to solve. Nonlinear equations cannot be solved in this way.
Nonlinear equations have not only a solution but an impossible one as well. This may seem odd, since the solution is supposed to be true!
But don’t let that scare you off. The solutions of nonlinear equations can be very cleverly designed, making them interesting to look at!
This article will talk about the different types of differential equation and how researchers solve them.
Homogeneous equations
When there is a change in the data that changes the equation, we call it a differential equation. In these cases, we must use more advanced techniques to solve the equations.
In most cases, when there is not a change in the data, then there is no need for solutions. However, when there is a change in one of the variables, then there may be a solution.
Sometimes solutions have special names such as stable or nonlinear equations. Other times, solutions just don’t work and we have to consider them illegal. In this case, we consider why it did not work and what could cause it to work again.
Non-homogeneous equations
Most linear algebra problems have a well-defined solution, and we can rely on that for our non-linear problem. However, in some cases, the solution does not exist because of the equation.
In these cases, we have to create a solution based on the known variables. This can be done by introducing new constants or variables or taking existing ones away.
This is what happens when we do not consider the differential equation to be differentiable. Then, we cannot know if it has a slope or not because there would be an change in it.
It is important to note that this kind of problem cannot be solved using just one variable alone! We have to use both to find the problem’s solution.
Example of a differential equation
A very common equation that changes from place to place is the differential equation: dy/dx=e^y(3x^2-6x). This is the classic example of a differential equation, and it’s pretty easy to spot.
When there’s a change in scale in the equation, such as when x increases or decreases, then there may be a new problem posed. For example, when 3x increases, that may mean that 6x decreases.
Since the scale of the problem does not change much, there’s not much need for complicated rules for solving differential equations.
Solution methods for differential equations
In fact, there are several ways to solve differential equations. One method is called solution methods. Another is called integration methods. Both have their place.
Solution methods use tools such as chains, grid systems, or approximation algorithms to find a solution. They can be used linear or nonlinear equations, and simple or complex!
Integration Methods use a concept known as integration. This can be done using various techniques such as Integration by Leibniz, Newton’s Method, or Even-Toh Method. All of these rely on the assumption that the equation has a variable range!
Chain/grid methodologies require you to develop mental models of how different parts of the system work in order to determine how they affect the system as a whole. These must be ‘put on the ground’ before being used.
Exact solutions for simple cases
For cases where the change in x is very small, or where y is constant, there is a simple solution. Just solve for x and for y and you’re good to go.
If x = 1, then dy/dx = 1 which is the same value as dx/dy = 1. If x = 2, then dy/dx = 2 which is the same value as 2x+2=2 which has already been solved for in your problem.
If y = 2x+3, then dy/dy = 4 which is the same value as 4xy+4=4 which has already been solved for in your problem.
Linear homogeneous equations with constant coefficients
Linear homogeneous equations with constant coefficients are a common type of equation. These types of equations don’t change the value of an element, but instead shifts the location of an element.
This type of equation can be tricky, as some are not. These include linear equations that have no variable and only one constant coefficient.
These variables must be substituted into the equation, and then the coefficient can be found. This is tedious and time consuming, so it is good to know how to solve linear homogeneous equations that have no variable.
There are two ways to solve a linear homogeneous equation with no variable. One way is to create another variable and change the function being solved for.
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