Linear Equation: Definition & Meaning

[Maxwell]

What does it mean if an equation is linear?

Like what is a linear differential equation?

 

[JasonRox]

A linear equation is a straight line where m is defined. (m is the slope)

It is in the form y=mx+b, where m is the slope and b is the y-intercept.

This second part is a guess.

A linear differential equation I "THINK" is a polynomial to the second degree.

I think this because the derivative of:
$$y=Nx^2+Mx+W$$
is
$$\frac{dy}{dx}=2Nx+M$$

...which is linear.

2N is the slope of the derivative and M is the y-intercept.

 

[cepheid]

No.

Remember that a differential equation is an equation that contains a function y and one or more of its derivatives. The equation need only be linear in y, but not necessarily in the independent variable (x). The general form of a linear first-order differential equation is:

$$\frac{dy}{dx} + p(x)y = g(x)$$

Where $p(x) \ \ \text{and} \ \ g(x)$ are functions of x. So if I write:

$$\frac{dy}{dx} + (\cos{x})y = e^x$$

I guarantee you that the solution is not a second-degree polynomial in x! I seem to remember there being a great tutorial thread on D.E.'s in that subforum.

Jason, I guess you can think of the equation you wrote as the simplest case of a linear first order d.e., in which p(x) = 0. So we're left with:

$$\frac{dy}{dx} = g(x)$$

The solution can be found simply by integrating in this case. Not so simple with the previous example.

 

[HallsofIvy]

The crucial point about a "linear differential equation", indeed about "linear" problems in general, is that we can combine two solutions to make a third solution.

If y₁ and y₂ both satisfy the equation a(x)y"+ b(x)y'+ c(x)y= 0 then any "linear combination" of them, py₁+ qy₂, where p and q are numbers, does also:
a(x)(py₁+ qy₂)"+ b(x)(py₁+ qy₂)'+ c(x)(py₁+ qy₂)=
a(x)py₁"+ a(x)qy₂"+b(x)py₁'+ b(x)qy₂'+ c(x)py₁+ c(x)qy<sub>2[/sub=

p(a(x)y1"+ b(x)y1'+ c(x)y1)+ q(a(x)y2"+ b(x)y2'+ c(x)y2=

p(0)+ q(0)= 0.

Similarly, if f(x)= ax, then f(nx+my)= a(nx+my)= n(ax)+ m(ay)= nf(x)+ mf(y).

If, however, f(x)= x², then f(x+y)= (x+ y)²= x²+ 2xy+ y². The fact that f is NOT linear means that that term 2xy in which the two solutions x and y "interfere" with one another.</sub>