Linear equation and linear differential equations. General form of linear equation: Ax+By+C=0. Slope intercept form: y=mx+b. Is this also true for linear differential equations?

Altenbraknz 2022-09-27 Answered
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:
A x + B y + C = 0
Slope intercept form:
y = m x + 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."
d y d x = P y + 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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Answers (1)

acorazarxf
Answered 2022-09-28 Author has 9 answers
Step 1
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.
Step 2
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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