# Linear or Non-Linear Differential Equations

1. Dec 20, 2012

### msell2

(d4x)/(dt4) + (1/(1+t))*(d2)/(dt2) = x(t)
Is this differential equation linear or non-linear? I don't understand the difference.

2. Dec 20, 2012

### pasmith

$$\frac{\mathrm{d}^4x}{\mathrm{d}t^4} + \frac1{1+t} \frac{\mathrm{d}^2x}{\mathrm{d}t^2} = x$$
is linear, because it can be written in the form
$$a_0(t) x + a_1(t) \frac{\mathrm{d}x}{\mathrm{d}t} + a_2(t) \frac{\mathrm{d}^2x}{\mathrm{d}t^2} + \dots + a_n(t) \frac{\mathrm{d}^nx}{\mathrm{d}t^n} = f(t)$$
for given functions $a_k(t)$ and $f(t)$. It does not, however, have constant coefficients.

3. Dec 20, 2012

### HallsofIvy

Staff Emeritus
A differential equation is "linear" as long as there are no functions of the dependent variable, here x, or its derivatives, other than just the usual "linear" functions, multiply or divide by a number and add or subtract.

In particular, that $d^4x/dt^4$ is just the fourth derivative. Had it been $(dx/dt)^4$, the first derivative to the fourth power, then the equation would have been non-linear.