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Calculus

What is a Differential Equation

Differential equations are powerful equations that are often used to describe real-world phenomena, like population growth. Often, differential equations cannot be solved explicitly, but approximations give us enough information to make conclusions. 

Differential equations are characterized by two key parts that separate them from other equations: they include a function (ex. \(y\)) and the derivatives of that function (ex. \(y', y'', \) etc.). 

Example

The following are all differential equations: 

  1. \(y + y' = 7x\)
  2. \(y = y'\)
  3. \(\frac{dP}{dt} = kP\) (Population Model)

The solution of a differential equation is the function \(y\) that satisfies the equation. There will often be more than one solution to a differential equation.

Example

Verify that \(y=x-x^{-1}\) is a solution to the differential equation \(xy' +y = 2x\).

Solution

Our differential equation includes \(y\) and \(y'\). We have \(y\), so lets compute \(y'\): 

\(\frac{d}{dx}\left( x-x^{-1}\right) = 1-(-x^{-2}) = 1+x^{-2}\)

Now, we can plug these equations for \(y\) and \(y'\) in the left hand side of the differential equation: 

\(xy'+y = x(1+x^{-2})+(x-x^{-1}) = x+x^{-1} + x-x^{-1} = 2x\)

which is equivalent to the right-hand side. Thus, \(y=x-x^{-1}\) is a solution to \(xy' +y = 2x\). 

In a future lesson, we'll look at different solving techniques. 

Classification of Differential Equations

There are different ways to classify and describe differential equations. Classification of differential equations can help narrow down solving techniques. 

1. Order

The order of a differential equation is the order of the highest derivative that occurs in the equation. For example, in the differential equation \(y' = xy^3\), the order is 1. In \(y' = 2y'' + 4x\), the order is 2. In the second example, we consider the \(y''\) and ignore the lower-order derivative. The first example is called a first-order differential equation. The second example is a second-order differential equation. 

Examples

Classify these differential equations based on order:  

  1. \(\frac{d^2y}{dx^2}-x^3= y+3\)
  2. \(\frac{d^2y}{dx^2} +y = 0\)
  3. \(y'-sin(x) = 0\)

Solution

  1. This is a second-order differential equation.
  2. This is a second-order differential equation.
  3. This is a first-order differential equation.
2. Degree

The degree of a differential equation is the exponent on the highest-order derivative. For example, looking at \(y' = 6x+2\), we see that the highest order derivative is \(y'\). The exponent on \(y'\) is 1. Thus, this is a degree one differential equation. The differential equation \((y')^2 + y = x^2\) would be called a degree two differential equation since the exponent on the highest derivative, \(y'\), is 2. 

Examples

Classify these differential equations based on their degree: 

  1. \((\frac{d^2y}{dx^2})^3 - (\frac{dy}{dx})^2 = 1\)
  2. \((y'')^2 + 3(y')^3 = 3x^5\)
  3. \(2y' - 3y'' = 4x^3+2\)

Solution

  1. This is a degree 3 differential equation. 
  2. This is a degree 2 differential equation. (Note: the exponent of 3 on the \(y'\) does not count since it's not the highest order derivative of y)
  3. This is a degree 1 differential equation. 
3. Other Classifications

Differential equations can be either ordinary or partial. 

a) Ordinary Differential Equations (ODEs) have only one independent variable in the function. All the above examples are ordinary differential equations. 

b) Partial Differential Equations (PDEs) have functions with more than one variable. The derivatives are then partial derivatives. 

Additionally, differential equations can be linear. A linear differential equation is of the form \(\frac{dy}{dx} + P(x)y = Q(x)\) where \(P(x), Q(x)\) are functions of x. Note that there are no functions of \(y\) (like \(y^2, y^3, \sqrt{y}\), etc). 

Furthermore, differential equations can be separable. This means the differential equation is of the form \(\frac{dy}{dx} = g(x)f(y)\). Essentially, the right side of the function can be factored into two separate parts, one a function of x and the other a function of y. Note that this is a first-order differential equation (only first derivatives).

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