Definition of Linear Differential Equation

I am a 13 year old self teaching myself Differential Equations from a website and a book, I came across the definition of a Linear Differential Equation but I didn't understand the definition, I looked on other websites and even found a Chinese book from the library about differential equations (I'm Chinese just saying) but I still didn't understand the definition of it, I think its just the Notation messing me up. Can someone explain to me what exactly a Linear Differential Equation is in simple words?

I've seen a definition that looks like this: $$a_n(t)y^{(n)}(t)+a_{n-1}(t)y^{(n-1)}(t)+......+a_1(t)y^{(1)}(t)+a_0(t)y(t)=g(t)$$

Yan Yau

• Where do we have to start? Do you know what a derivative is? – Gerry Myerson Oct 3 '13 at 9:53
• yes, I know what A derivative is, I know Differential and Integral Calculus – YYC Oct 3 '13 at 9:54

Yan Yau. I'm pretty happy to see you like this. Linear differential equation is a kind of differential equations such that if you regard derivatives (not matter of first order or higher order) as $x,y,...$, then there's no term like $x^3$ or $xy$ etc.

• Come on, Yan Yau. Keep enthusiasm and passion! – Shuchang Oct 3 '13 at 10:11

Let's write $y^{(n)}$ for ${d^ny\over dx^n}$. A linear differential equation is an equation of the form $$a_ny^{(n)}+a_{n-1}y^{(n-1)}+\cdots+a_1y'+a_0y=f(x)$$ where $a_0,a_1,\dots,a_n$ are given functions of $x$.

A linear differential equation is a linear combination of derivatives, e.g.

$$f(x) y'''(x) + g(x) y''(x) + h(x) y'(x) + s(x) y(x) + t(x) = 0.$$

Of course a general differential equation could have terms that depend on the derivatives in a nonlinear way, for instance:

$$y(x) - y'(x)^2 = 0$$

or

$$f(x) y(x) y''(x) + g(x) = 0$$

These are examples of nonlinear differential equations.