A bit about differential equations

[Luongo]

For first order linear differential equations when is it alright to use the
dy/y-b/a = -adt form and when you must use the integration factor technique? In general i was able to obtain a solution using both methods.

Also, how do i draw a direction field which contains both t and y variables without solving and graphing the differential equation? i can only obtain the direction field at y = 0, it becomes too complicated if t is not 0, does anyone have any tips for drawing direction fields where y,t are variables for y'?

 

[HallsofIvy]

For first order linear differential equations when is it alright to use the
dy/y-b/a = -adt form and when you must use the integration factor technique? In general i was able to obtain a solution using both methods.

You use whatever method is easiest. There always exist an integrating factor but, except for linear equations, it may be very difficult to find. As for "dy/y- b/a= -adt", I don't know what you mean. It is impossible to have a differential equation of that form. If any term of a differential equation has the differentials, dy and dt, every term must.

Also, how do i draw a direction field which contains both t and y variables without solving and graphing the differential equation? i can only obtain the direction field at y = 0, it becomes too complicated if t is not 0, does anyone have any tips for drawing direction fields where y,t are variables for y'?

If you are given dy/dt= f(t,y) then drawing the direction field at (t,y) is just a matter of evaluating f(t,y). Why should that be any easier at y= 0?