Linear equation and linear differential equations

I remember noting from an algebra class that $x$ and $y$ of a linear equation neither divide or multiply with each other which is somewhat clear from the forms of linear equations:

General form of linear equation:

$Ax + By + C = 0$

Slope intercept form:

$y = mx + b$

Is this also true for linear differential equations?

The definition goes like this: "A differential equation is said to be linear if the dependent variable and its differential coeficients (derivates) occur only in the first degree and not multiplied together."

${dy \over dx} = {Py + Q}$

Where P, Q are functions of $x$ only. What exactly does this mean?

Does the algebraic linear equation has something to do with linear differential equation?

• I may be wrong, but if we define $L$ to be the operator for the differential equation(in your case $L = \dfrac{d}{dx} - P$), and $L$ is linear, then the differential equation $Lu = f$ is said to be linear. $L$ is said to be linear if you have constants $a$ and $b$ and functions $x$ and $y$, then $L(ax+by) = aL(x)+bL(y)$. Sep 3 '15 at 23:42
• It sounds like it is saying about the dependent variable y in the case ${dy \over dx} = {Py + Q}$ not to be multiplied with ${dy \over dx}$ itself. Right?
– cpx
Sep 3 '15 at 23:56
• that's one way to stay linear, yes. Sep 4 '15 at 2:58