Linear Differential Equation

What are Linear Differential Equations?

A differential equation is an equation for an unknown function, where its derivatives are involved. But what does it make it linear?

You can say that a differential equation is linear if each dependent variable appears in linear fashion. This means that the dependent variable and/or its derivatives are all raised to the power of \(1\), they are not multiplied together, and they are not part of a special function, like a trigonometric function or the exponential function.

Despite the above considerations, the independent variable can be non-linear. That is, the independent variable can be squared, as part of the sine function, and so on.

Below are some examples of linear differential equations.

\[ \frac{\mathrm{d}y}{\mathrm{d}x}+2y=0\]

\[ x^2\frac{\mathrm{d}^2y}{\mathrm{d}x^2} +x\frac{\mathrm{d}y}{\mathrm{d}x}+4y=e^x\]

\[ \frac{\mathrm{d}y}{\mathrm{d}x}+\left(\cos{x}\right)y=x^2\]

Linear vs Nonlinear Differential Equations

Just like there are linear differential equations, there are also nonlinear differential equations.

A nonlinear differential equation is a differential equation that is not a linear differential equation.

Simple, right? This means that the dependent variable and/or its derivatives are either:

• Being multiplied by each other

• Are raised to a power other than \(1\)

• Are part of a special function, like a trigonometric function or exponential function

Here are some examples of nonlinear differential equations.

\[ \frac{\mathrm{d}y}{\mathrm{d}x}+xy^2=0\]

\[ y\frac{\mathrm{d}^2y}{\mathrm{d}x^2}+xy=\ln{x}\]

\[ \frac{\mathrm{d}y}{\mathrm{d}x}+\sin{y}=\cos{x}\]

Linear differential equations can be used to describe some natural phenomena, including but not limited to:

• Electromagnetic waves.

• The diffusion of heat.

• Electronic circuits.

• Oscillatory movement, like springs and pendulums.

Meanwhile, nonlinear differential equations can describe things like:

• Weather.

• Fluid dynamics.

• Population dynamics.

First Order Linear Differential Equations

Differential equations are usually classified based on their order. A first order linear differential equation is a linear differential equation where the highest derivative involved is a first derivative. A first order linear differential equation can always be written in the form

\[ \frac{\mathrm{d}y}{\mathrm{d}x} + P(x)y=Q(x).\]

This is known as the standard form of a first order linear differential equation.

You might also come across differential equations written in prime notation, so the above differential equation can also be written as

\[ y'+P(x)y=Q(x),\]

where the dependence of \(y\) is not explicitly stated, but has to be assumed according to the context.

Note that you might be given a linear differential equation that is not written in either way. In these cases you need to do some algebra to rewrite it.

Linear Differential Equations with Constant Coefficients

You have seen that a first order linear differential equation can always be written as

\[ \frac{\mathrm{d}y}{\mathrm{d}x}+P(x)y=Q(x).\]

This means that, in general, \( P \) and \( Q \) are functions of \(x.\) However, in the special case that the functions \( P(x) \) and \( Q(x) \) are constant functions, you have a linear differential equation with constant coefficients.

The following differential equations are all linear differential equations with constant coefficients.

\[ \frac{\mathrm{d}y}{\mathrm{d}x}+3y=0\]

\[ \frac{\mathrm{d}y}{\mathrm{d}x}-\pi y=4\]

\[ \frac{\mathrm{d}^2y}{\mathrm{d}x^2}-5\frac{\mathrm{d}y}{\mathrm{d}x}-y=e\]

You can identify a linear differential equation with constant coefficients by noting that the independent variable does not appear explicitly.

Solving first order linear differential equations with constant coefficients is a straightforward task as they are Separable Equations. Consider the differential equation

\[ \frac{\mathrm{d}y}{\mathrm{d}x}+ay=b.\]

The above equation can be separated. First isolate the term containing the derivative, so

\[ \frac{\mathrm{d}y}{\mathrm{d}x} = b-ay.\]

The right-hand side of the equation can be seen as a function of \(y,\) that is

\[ g(y) = b-ay,\]

so

\[ \frac{\mathrm{d}y}{\mathrm{d}x} = g(y).\]

This tells you that the differential equation can be solved by separation of variables. Begin by rewriting the differential equation in terms of the differentials of \( x \) and \( y,\) that is

\[ \mathrm{d}y = (b-ay)\,\mathrm{d}x,\]

and then divide both sides of the equation by \( b-ay,\) obtaining

\[\frac{1}{b-ay} \, \mathrm{d}y = \mathrm{d}x.\]

From here you can integrate both sides. The left-hand side is one of the Integrals Involving Logarithmic functions, so

\[-\frac{1}{a} \ln{(b-ay)} = \int \mathrm{d}x,\]

and the integral of a differential is just the variable of integration, this way you can write

\[-\frac{1}{a} \ln{(b-ay)} = x+C.\]

Next, you need to isolate \( y,\) so

\[ \begin{align} \ln{(b-ay)} &= -ax-aC \\ b-ay &= e^{-ax-aC} \\ ay &= b-e^{-ax-aC} \\ y &= \frac{b}{a}-\frac{1}{a}e^{-ax-aC}. \end{align}\]

This will nook nicer by using properties of exponents and some algebra to write

\[-\frac{1}{a}e^{-ax-aC} = e^{-ax}\left(-\frac{1}{a}e^{-aC}\right),\]

where \( -\frac{1}{a}e^{-aC} \) is still a constant as a whole, so you can rename it, lets say as \(A.\) This way

\[y=Ae^{-ax}+\frac{b}{a}\]

is the general solution for a first order linear differential equation with constant coefficients.

Formula for Solving Linear Differential Equations

Usually, there are no formulas for solving differential equations. Luckily, in the case of first order linear differential equations, you can obtain a formula by using what is known as an integrating factor.

Consider a first order linear differential equation written in standard form, that is

\[ \frac{\mathrm{d}y}{\mathrm{d}x}+P(x)y=Q(x).\]

The general solution to the above differential equation is

\[ y = \frac{1}{\alpha (x)} \left( \int \alpha(x) \, Q(x) \, \mathrm{d}x + C \right),\]

where

\[ \alpha(x)=e^{\int P(x)\,\mathrm{d}x}\]

is known as an integrating factor.

Just like indefinite integrals, differential equations have families of solutions. By plugging different values of the integration constant, \(C,\) you get different solutions to the differential equation.

You can follow these steps in order to use the formula for solving first order linear differential equations that are written in standard form:

1. Calculate \( \int P(x) \, \mathrm{d}x.\) There is no need to add an integration constant in this step!

2. Find the integrating factor \( \alpha(x).\) This can be done by plugging in \( \int P(x) \, \mathrm{d}x \) into the exponential, that is,

   \[ \alpha(x) = e^{\int P(x) \, \mathrm{d}x}.\]

3. Calculate \( \int \alpha(x)\,Q(x)\,\mathrm{d}x.\)

4. Use the formula for the general solution of a first order linear differential equation.

You can look at some examples in the following section.

Linear Differential Equation Examples

The steps for solving a linear differential equation are better understood with examples. Let's dig in!

Calculus is all about practice! Here is another example.

How about one with constant coefficients?