Linear or Non-Linear Differential Equations

[msell2]

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

 

[pasmith]

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

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

 

[HallsofIvy]

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.