First Order Differentials Flashcards
What does rate of change of some variable become
D whatever / dt
How to do separation of both variables
Remember separate variables, try keep y on left side with no other number only
Now you can integrate both sides as equation
Both CONSTANTS of integration add together and just add once on the right
How to manipulate exponential or logs to make it neater
What to MAKE SURE TO SAY
If you have e to the c, remember that’s just multiplying e c by the thing, and if you say let e to the c = A. MUST SAY , then good
For logs, can convert c into log A, and then use log rules due to additon to multiply
Every time you app,y operation or change c, what must do
Introduce a NEW VARIBALE
How to always try leave solved differential equation
Try leave in terms of y
If wacky y on one side, what to do
Just factorise for y like you were solving for y, need a way to get ALL THE y on one side and x too, that you can multiply it when ever
When can you not use integration by parts?
When there is a function in a function! So e to the x sqaured, you won’t be able to use parts
Thus use reverse chain rule, substitiojn
In General for recap integration, you need to divide by the DERRIVSTIVE of whatever inner function is whenever reversing chain rule! And can only successfully do this if the derrivstive CLEANLY divides by something in numerator, so in those cases you can integrate something like sin2 . However when you can’t, need to cknvert this to integrate!
What is Limitation of the cooling
Is that the temp on infinite will never reach the temp of surrounding, due to the nature if exponential?
Rememebr if it’s a rate of DECREADE, what to do?
Must make it - D whatver / dy, and then multiply - to the other side
What makes it 1st order or 2nd order differential?
The higher power of the dy / dx
What is x dot y dot, specifically
Anything differentiated with RESPECT TO TIME, BECOMES x dot
How should you go about trying to solve every differential first
Chekc you can COMPRESSS, and that it’s a perfect integral first!
- if not then make it compressed, differentiate and see what the factor is off by, then adjust accordingly
This is a dodgy method but it works for dodgy qs and normally it’s when you can FACTOR Y OUT THE Q, then it will be sticky
Otherwise if can’t take y out, then Integrating fsctor! Remember to get into the form
Remember if there is y in the equation for 1st how to do
Find perfect derivative differntiate and adjust the perfect derrivstive accordingly!
Won’t be clear cut
How to check if perfect derrivstive?
If the function of y is the differential of the function of the dy by dx it is
Remmeber in questions where modelling velocity etc, how can you find c maybe?
Check moving from rest statement or something, this means when t = 0 v =0!
