Core Pure - Differential Equations Flashcards

1
Q

What is a homogenous differential equation?

A

A differential equation that is equal to 0

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2
Q

What are the 4 types of first order differential equations?

A

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

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3
Q

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

A

Just dy/dy

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4
Q

How does separating variables work?

A

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.

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5
Q

What may be common to do when separating variables?

A

Add 2 fractions together or factorise

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6
Q

What is exact form?

A

You realise that it is just a product rule

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7
Q

What si the integrating factor?

A
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8
Q

When can you use integrating factor?

A
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9
Q

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

A

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

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10
Q

How does integrating factor work?

A

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)

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11
Q

What does non homogeneous mean?

A

Ot is not equal to 0

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12
Q

How do you solve a homogeneous second order differential equation?

A

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

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13
Q

What is the auxiliary equation (aux:)

A

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

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14
Q

What are the 3 different cases of the aux:?

A

2 real distinct roots
Repeated real root
Complex conjugate solutions

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15
Q

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

A
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16
Q

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

A
17
Q

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

A
18
Q

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

A

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

19
Q

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

A

De moivre’s

20
Q

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

A

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

21
Q

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

A

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

22
Q

How do we form the PI, in general?

A

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

23
Q

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

A

Exponentials
Algebra/polynomials
Trigonometry

24
Q

Explain the PI method?

A

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)

25
Q

What is the general PI for exponentials?

A
26
Q

What is the general PI for polynomials?

A

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

27
Q

What is the general PI for trigonometry

A
28
Q

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

A

On the end of 1 side of the inequality

29
Q

What are some tricky PI’s?

A