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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?

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  • $\begingroup$ 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 $L(ax+by) = aL(x)+bL(y)$. $\endgroup$
    – DaveNine
    Sep 3 '15 at 23:42
  • $\begingroup$ 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? $\endgroup$
    – cpx
    Sep 3 '15 at 23:56
  • $\begingroup$ that's one way to stay linear, yes. $\endgroup$
    – DaveNine
    Sep 4 '15 at 2:58
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It's saying that if x and y (or their derivatives) are multiplied together in any way, it's not considered a linear differential equation because it's not solvable in the usual ways that linear ODE's are.

This relates to normal linear equations in that if you have an equation where x and y are multiplied or otherwise modify each other in a way that prevents strict separation in the polynomial, they do not have a linear relationship. For example, the plot of y=1/x is not a line while y=x is.

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