Equations are statements in which two expressions are equal to each other. If these equations consist of derivates of the dependent variable \(y\), it is a differential equation (DE).
Some examples of differential equations are the following:
- \(y^{\prime \prime}+y^{\prime}+2 x y=0\)
- \(\frac{d y}{d x}+x y=y\)
- \(\frac{d^3 y}{d x^3}-\frac{d^2 y}{d x^2}+y=x\)
Differential equations have lots of uses. For example:
- We can model growth and decay
- We use it to explain Newton's law of cooling
- We use it to model structure deflections
Bottomline, we use these equations to represent specific real-world events.
Order of a Differential Equation
Every differential equation has an order. It is the highest derivative order in the equation. To illustrate, let's use the differential equations above and find their order:
- In the DE \(y^{\prime \prime}+y^{\prime}+2 x y=0\), its order is 2.
- In the DE \(\frac{d y}{d x}+x y=y\), its order is 1.
- In the DE \(\frac{d^3 y}{d x^3}-\frac{d^2 y}{d x^2}+y=x\), its order is 3.
Summary
A differential equation (DE) is an equation that consists of derivates of the dependent variable \(y\).
Differential equations have many uses, and we use them to model real-world events.
Every differential equation has an order. It is the highest derivative order in the equation.
