Core Pure - Differential Equations Flashcards
What is a homogenous differential equation?
A differential equation that is equal to 0
What are the 4 types of first order differential equations?
Very simple (just dx/dy = 1 variable)
Separating variables
Exact form
Integrating factor
Which form will never usually come up and may be mistaken easily?
Just dy/dy
How does separating variables work?
You multiply out the dy/dx to get 2 integrals, then get all of g(y) on on 1 side and the f(x) on the other side.
What may be common to do when separating variables?
Add 2 fractions together or factorise
What is exact form?
You realise that it is just a product rule
What si the integrating factor?
When can you use integrating factor?
For integrating factor to work what also needs to be the case?
The dy/dx needs to have a coefficient of 1
(So divide by the coefficient)
How does integrating factor work?
You recognise that it isn’t exact for or splitting the variables, then you find the integrating factor, R, then multiply everything by R and that will be exact form (if not then you did it wrong)
What does non homogeneous mean?
Ot is not equal to 0
How do you solve a homogeneous second order differential equation?
Find the auxiliary equation
Solve it
Write it in the right form
What is the auxiliary equation (aux:)
Replace dydx with m
Eg m^2 +2m+1
What are the 3 different cases of the aux:?
2 real distinct roots
Repeated real root
Complex conjugate solutions
What do you do to solve a homogenous second order differential equation when the aux: is 2 distinct real roots?
What do you do to solve a homogenous second order differential equation when the aux: is 2 real repeated roots?
What do you do to solve a homogenous second order differential equation when the aux: is complex conjugates
Why do we see a lot of e^x when we do second order differential equations?
We know the answer will a a function which, when differentiated once and twice will sum to be 0. e^x is a good candidate for this
Why does the case with complex conjugates have cos and sin in it?
De moivre’s
What is the method to solve non homogenous second order differential equations?
Solve it as if it was a homogenous second order differential equation
Chose a suitable Particular Integral (PI)
CF + PI
Explain partially the theory behind this method for solving non homogenous second order differential equations?
The CF makes it =0 to we can add a function (the PI) to get it to the answer
How do we form the PI, in general?
We recognise a form then we make it as general as possible
What are the 3 different cases that you need to be able to solve with a PI?
Exponentials
Algebra/polynomials
Trigonometry
Explain the PI method?
You create a generalised PI then you differentiate it 2 and substitute it into the differential equation and solve for the unknowns (the equal sign will term into an identical sign)
What is the general PI for exponentials?
What is the general PI for polynomials?
You do it to the highest power:
If it is only a constant then you do
Y=a
Dy/dx … = 0
What is the general PI for trigonometry
When integrating first order differential equations, where can you add the c?
On the end of 1 side of the inequality
What are some tricky PI’s?
