Question

Which of the following can be put in the form of a linear differential equation.



a)dy/dx = (10x^2 + 6x + 5)y + 10x + 6


b) dy/dx = 5x^7 + 7x^6y + 7x^2y + 9x + y + 5

c) (tan(x)) dy/dx = x^3y + 3

Answer 1

@primeralph @Psymon @John_ES

Answer 2

Well, a linear equation is of the form:

\[\frac{ dy }{ dx }+ P(x)y = Q(x)\]So basically a derivative of y term, a regular y term and then some other function, be it a variable, constant value, etc. Of course we also have to be linear, as in no powers of y higher than 1. So from the looks of it, we might be able to get all of these to work into some sort of linear form
For a.

dy/dx - (10x^2+6x+5)y = 10x + 6

Nothing wrong with that one. We have y' + something y = something. Works out. For b.

dy/dx -(7x^6+7x^2+1)y = 9x + 5

Again, may look long and funky, but still is y' + something y = something. Also checks out
For the last one, this one will checlk out, too, we just cant have anything paired with dy/dx, we need that by itself. So we can do that and get:

dy/dx - (x^3cotx)y = 3cotx

So as far as I see it, all 3 work out.

Answer 3

Thanks