Week 5&6 2nd ODE

Question 1

What’s the form of a linear 2nd order ODE?

Answer 1

Question 2

how do you know if a linear 2nd order ODE is homogenous or non-homogenous?

Answer 2

r(t) = 0 - homogenous

Question 3

what else would be another solution?

Question 4

what’s the general homogenous 2nd order ODE form?

Answer 4

Question 5

what are the different cases for the roots for solving general homogenous 2nd order ODE’s?

Definitions?

Answer 5

1- real roots & different,

2- real roots & same,

3- complex roots

Question 6

How would you solve a general homogenous 2nd order ODE form?

Answer 6

1) solve the auxiliary equation

2) establish value of m

3) sub in appropriate form of the roots
(1- real roots & different,
2- real roots & same,
3- complex roots)

Question 7

What’s the solution to real roots & different cases

Answer 7

NOTE :
for dy/dx

Question 8

Solve this 2nd order ODE

Answer 8

1) use auxiliary equation
-compare a,b,c values

(CHECK DISCRIMINANT TO CONFIRM ROOT TYPE)

2) factorise and find roots

3) identify appropriate form
e.g. here m1 doesn’t = m2
so it’s real roots&different

4) use corresponding equation form
(use real roots&different one)
and sub m values into there

Question 9

What’s the solution to real roots & same cases

Answer 9

NOTE :
for dy/dx

Question 10

Solve this 2nd order ODE

Answer 10

1) use auxiliary equation
-compare a,b,c values

(CHECK DISCRIMINANT TO CONFIRM ROOT TYPE)

2) factorise and find roots

3) identify appropriate form
e.g. here m1 = m2
so it’s real roots&same

4) use corresponding equation form
(use real roots&same one)
and sub m values into there

Question 11

What’s the solution to complex roots cases

Answer 11

NOTE :
for dy/dx

Question 12

Solve this 2nd order ODE

Answer 12

A
1) use auxiliary equation
-compare a,b,c values

(CHECK DISCRIMINANT TO CONFIRM ROOT TYPE)

2) factorise and find roots
- have to use quadratic equation here to find complex roots

3) simplify roots

4) match the real and imaginary roots, where imaginary root has square root, real doesn’t

• obtain alpha, beta value

4) use corresponding equation form
(use complex roots one)
and sub alpha, beta values into there

Question 13

quadratic equation?

Answer 13

Question 14

What’s the complex root form?

Answer 14

a= real

b= imaginary

Question 15

match the real/ imaginary number

Answer 15

1) simplify

2)
real (alpha) = no square root

imaginary (beta) = square root

Question 16

What’s the discriminant?

Answer 16

Question 17

How do you solve 2nd order non-homogenous ODE’s?

Answer 17

Question 18

How do you find the particular solution ? (for 2nd order non-homogenous ODE’s)

Answer 18

1) assume general form of the function on RHS of equation,
- compare the RHS with TABLE

2) Differentiate twice

3) sub this in the main equation

4) collect like terms/ simplify

5) equate the coefficients/ find the coefficient values

6) sub back into yp equation

yp = particular solution equation

Question 19

particular solution for non-homogenous ODE TABLE

Answer 19

Question 20

what are the rules for the particular solution table?

Answer 20

1) basic rule

2) modification rule

3) summation rule

Question 21

what is the basic rule (particular solution table)

Answer 21

If the RHS of ODE matches with a function in first column, sub in the corresponding value in the second column on same row

• find unknown coefficients by differentiating and sub into ODE

Question 22

What is the multiplicity?

For 2 distinct roots?
For 1 repeated root?

Answer 22

power of your linear (one) factor

Question 23

what is the modification rule (particular solution table)

Answer 23

• If the particular solution is a solution of the homogenous equation (where you solve as if it’s homogenous):

multiply the particular solution by (independent variable)^m,

(independent variable= BOTTOM)
where m= multiplicity of the root of the auxiliary equation

Question 24

what is the summation rule (particular solution table)

Answer 24

if RHS is a sum of terms in the first column of the TABLE,

For the particular solution use the sum of the functions in the corresponding rows

Question 25

Solve this 2nd order ODE

state if its homogenous or not

Answer 25

non-homogenous (as RHS doesn’t = 0)

1) Solve equation as if 2nd order ODE (=0),
Get the homogenous solution

2) Find the particular solution
(use table to find corresponding value for RHS)

3) Combine the 2 equations together

Question 26

Solve this 2nd order ODE

state if its homogenous or not

Answer 26

Non- doesn’t = 0
If you skipped the modification rule: after differentiating twice and subbing, particular solution = 0 on LHS and wouldn’t balance

1) solve as if it’s homogenous

2) find particular solution

3) combine equations to get general solution

4) Gives initial conditions, so find A and B using these prompts

Question 27

Solve this 2nd order ODE

state if its homogenous or not

Answer 27

Non- doesn’t = 0

Question 28

If the RHS in a 2nd order ODE = IMAGE

then what’s the particular solution?

Answer 28

Question 29

If the RHS in a 2nd order ODE = IMAGE

then what’s the particular solution?

Answer 29

Question 30

Answer 30

Question 31

Take a cantilever with free oscillations

what assumptions can be draw from this?

Answer 31

1) deflection of the load is proportional to the load
(not valid in breakage scenarios nor non-linear deflections)

2) assume mass of load is&raquo_space; the mass of the cantilever

3) ‘x’ is displacement of the load from equilibrium

Question 32

Solve this ODE

Answer 32

Question 33

hookes law?

Answer 33

F = ky

k = spring constant
y = displacement
f= spring force

Question 34

Answer 34