When dealing with differential equations (DE), our primary goal is finding a solution to the DE.
What is a Solution?
A solution is usually a function that would satisfy the DE. Aside from being a function, it is an expression free of math derivatives.
To illustrate, say we have a differential equation \(y^\prime – 2x = 0\). The solution to this equation is \(y = x^2 + C\). Don't worry about how we got it; we will discuss the different methods to solve DE soon.
We determine that a solution solves the DE if it correctly evaluates back to the equation when plugged in.
To illustrate, let's evaluate the differential equation \(y = x^2 + C\). To do that, we substitute \(y = x^2 + C\) and its derivates to the original expression:
- \(y^\prime – 2x = 0\)
- \(2x - 2x = 0\), the derivative of the solution \(y = x^2 + C\) is \(y' = 2x\)
- \(0 = 0\)
If the equality is correct, then the expression is a solution. In this case, it is.
Now that we know what a solution is. It's time to discuss the two types of solutions: (1) general and (2) particular.
General Solution
For every differential equation, there is a general solution. Returning to \(y' – 2x = 0\), we say that the solution \(y = x^2 + C\) is its general solution.
One characteristic of general solutions is always having this arbitrary constant \(C\). This constant is similar to the constant of integration. We call it the general solution because this constant can be of any value.
For example, we assume multiple values of \(C\): 0, 1, 2, 3, etc., to the solution curve \(y = x^2 + C\). The solutions will be:
- \(y = x^2\)
- \(y = x^2+1\)
- \(y = x^2+2\)
- \(y = x^2+3\)
Despite having different expressions, these functions are solutions to the differential equation. We use \(C\) as a general placeholder to hold all these possible solutions to the DE.
We can plot these functions on a graph to see the solution curves to the differential equation. We sometimes call this collection of graphs the solution family curves.
Particular Solution
We call a solution particular if we are interested in a specific answer among the general solution curves. Usually, this happens if we have a condition.
Consider the previous example; let's say we want a solution that satisfies the following criteria: variable \(y\) must be zero when \(x\) is 5.
If you solve for \(C\) using this information:
- \(y = x^2+C\)
- \(0 = 5^2+C\)
- \(C=-25\)
Substituting this \(C\) to the original solution, we'll have \(y = x^2-25\). This result is what we call a particular solution. It only exists if we have default values for its arbitrary constants.
To construct particular solutions, we must take a few notes:
- Conditions must consider a relationship between values. In our example, \(y = 0\) when \(x = 5\).
- The number of conditions must equal the number of arbitrary constants. For example, if you have three arbitrary constants, there must be three conditions.
Summary
When dealing with differential equations (DE), our primary goal is finding a solution to the DE.
A solution is usually a function that would satisfy the DE. Aside from being a function, it is an expression free of math derivatives.
We determine that a solution solves the DE if it correctly evaluates back to the equation when plugged in.
There are two types of solutions: general and particular.
For every differential equation, there is a general solution. One characteristic of general solutions is always having this arbitrary constant \(C\), which acts as a placeholder to hold all possible solutions to the DE.
We call a solution particular if we are interested in a specific answer among the general solution curves. Usually, this happens if we have a condition.
Conditions must consider a relationship between values in the equation.
The number of conditions must equal the number of arbitrary constants.
