Linear and non-linear differential equations

1. Aug 16, 2012

Daniel1992

I am not sure I understand Linear and non-linear differential equations properly so I will ask some question which someone will hopefully answer.

Is dx/dt = x + 2 a linear differential equation? If so does this mean that the rate of change of is constant?

Is dx/dt = x^2 + 4x a non linear differential equation?

2. Aug 17, 2012

NegativeDept

Linear ordinary differential equations are statements of the form:

linear combination of x(t) and its time derivatives = f(t)

The independent variable doesn't have to be called t, but it's a nice convention.

The first equation can be written:
$-x(t) + \frac{d}{dt}x(t) = 2$
The left side is a linear combination of $x$ and its first time derivative, and the right hand side is a (boring, constant) function of time. So it is a linear ODE.

In the second equation, the $x^2$ term prevents us from writing the ODE in the form linear combination of x and its derivatives = f(t). So it is nonlinear. For nonlinear ODEs, the superposition principle isn't guaranteed to work, and some other bad behaviors are allowed that would be impossible for linear systems. For example, I think your second equation grows hyperbolically: it blows up to ∞ in finite time.

3. Aug 17, 2012

Unstable

First, it is much easier for beginners to consider the dependencies of $t$ in the functions. So in your example $x$ is $x(t)$.

Try to write the differential equations in the following form:

$x’(t) = A(t)x(t) + g(t)$

If this is possible it is a linear differential equation.

In your first example: What is $A(t)$ and what is $g(t)$?

In the second example: Why is the above form not possible?