Year 2 - Differential Equations

Question 1

What is a homogenous differential equation?

Answer 1

A differential equation that is equal to 0

Question 2

What are the 4 types of first order differential equations?

Answer 2

Very simple (just dx/dy = 1 variable)
Separating variables
Exact form
Integrating factor

Question 3

Which form will never usually come up and may be mistaken easily?

Answer 3

Just dy/dy

Question 4

How does separating variables work?

Answer 4

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.

Question 5

What may be common to do when separating variables?

Answer 5

Add 2 fractions together or factorise

Question 6

What is exact form?

Answer 6

You realise that it is just a product rule

Question 7

What si the integrating factor?

Answer 7

Question 8

When can you use integrating factor?

Answer 8

Question 9

For integrating factor to work what also needs to be the case?

Answer 9

The dy/dx needs to have a coefficient of 1
(So divide by the coefficient)

Question 10

How does integrating factor work?

Answer 10

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)

Question 11

What does non homogeneous mean?

Answer 11

Ot is not equal to 0

Question 12

How do you solve a homogeneous second order differential equation?

Answer 12

Find the auxiliary equation
Solve it
Write it in the right form

Question 13

What is the auxiliary equation (aux:)

Answer 13

Replace dydx with m
Eg m^2 +2m+1

Question 14

What are the 3 different cases of the aux:?

Answer 14

2 real distinct roots
Repeated real root
Complex conjugate solutions

Question 15

What do you do to solve a homogenous second order differential equation when the aux: is 2 distinct real roots?

Answer 15

Question 16

What do you do to solve a homogenous second order differential equation when the aux: is 2 real repeated roots?

Answer 16

Question 17

What do you do to solve a homogenous second order differential equation when the aux: is complex conjugates

Answer 17

Question 18

Why do we see a lot of e^x when we do second order differential equations?

Answer 18

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

Question 19

Why does the case with complex conjugates have cos and sin in it?

Answer 19

De moivre’s

Question 20

What is the method to solve non homogenous second order differential equations?

Answer 20

Solve it as if it was a homogenous second order differential equation
Chose a suitable Particular Integral (PI)
CF + PI

Question 21

Explain partially the theory behind this method for solving non homogenous second order differential equations?

Answer 21

The CF makes it =0 to we can add a function (the PI) to get it to the answer

Question 22

How do we form the PI, in general?

Answer 22

We recognise a form then we make it as general as possible

Question 23

What are the 3 different cases that you need to be able to solve with a PI?

Answer 23

Exponentials
Algebra/polynomials
Trigonometry

Question 24

Explain the PI method?

Answer 24

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)

Question 25

What is the general PI for exponentials?

Answer 25

Question 26

What is the general PI for polynomials?

Answer 26

You do it to the highest power:
If it is only a constant then you do
Y=a
Dy/dx … = 0

Question 27

What is the general PI for trigonometry

Answer 27

Question 28

When integrating first order differential equations, where can you add the c?

Answer 28

On the end of 1 side of the inequality

Question 29

What are some tricky PI’s?