Second Order Differentiaks

Question 1

Why 2nd order

Answer 1

Because the value of derrivstive is 2nd

Question 2

How would we long method solve
- why can we add our two solutions at the end?

Answer 2

By assuming amdeer to equation, subbing this in by finding dy/dx and d2y/ dx2 and then solving to find m.

We find that we got two solutions, and subbing any of these in will give 0

• however adding them together ALSO works if we sub that in
• this is because they cancel independently and si adding them together you’re allowed

Thus we get ONE EQUATION with two constants

Question 3

So now that we saw way, how can we do short it (AE)

Answer 3

Auxhilhar equation means replace with m and solve for m

As long as you know the format it should be , then can find the answer very easily
- so repeated roots, distinct, real roots

Question 4

What are the THREE ways for distinct , repeated and complex?

Answer 4

Distinct = A e root x + B e root x

Repeated = (Ax+B) e to the root x

Complex = E to the real part (Acos (img ) x + B sin (img x)

Question 5

How do you get to the complex roots showing proof?

Answer 5

Find roots, and assume distinct so use first form

Now Solti up using indicted

So,it up again using Euler

Collect and manipulate to show

A and B must be resl here in our situations

Question 6

How does a linear differntiate using zu Hillary equation look

Answer 6

A e to the root of the x !

Question 7

How to do cyclic integration for integration by parts

Answer 7

Here, if yiu end up having to INTEGRATE what you initially wanted to integrated, let that be I, and add to other side which is already I, and now must divide !

Makes it very possible for parts

Question 8

How to solve 2 ODE non HOMOGENOUS?

Answer 8

Use left hand sind with AE to find first part of answer , called COMPLEMENTARY FUNCTION

Use right hand side with hints to guess the last part of solution, called PARTICULAR INTEGRAL

Question 9

How to go about finding the particular integral after learning each case?

Answer 9

Say let y = px +q

Then differntiate and sub into left

And COMPARE COEFFICIENTS EASY

Question 10

What are all the trial function scenarios

Answer 10

Constsnt = c
Linear = ax +b
Quadratic = a x 2 +bx +c
Expoential = ae ^ (same vlaue ) x
Trig = a sin (same) x + b cos (same x)

Sum of functions = sum of trial

SPECIAL CASE = x( in front of normal trial)

Question 11

What happens if special case AND repeated root?

Answer 11

That means you already have something like a x e to something x

So need a x SQAURED

(Multiply whatever you have by x basiclaly)

Question 12

So hot to fully solve 2ODE?

Answer 12

First auxhillary for complimentary, solve for complimentary

2) see if second part is of form of complementary. If it is, multiply by x
3) differentiate twice and sub in trial to oringak equation
4) compare coeeficntd and find the particular integral

Add them together done

Question 13

If there are two of the same function for particular ringegral what to do?

Answer 13

Still need to do SUM OF FUNCTIONS

Question 14

How to solve 2ODE particular solution, what to look out for to sub In?

Answer 14

If they say at time 0 the velocity is one or something , that means NEED to differntiate your equation and sub in

In General it is quite common for you having to differntiate and sub in somethinf for second , so if don’t know , then think what could the differential represent, oh speed, then sub in speed at that time etc…

Anyways you should get a DIMULATENOUS solution at the end!

Question 15

Why when a yute attached to a string extends the speed is given by change in extension over time

Answer 15

They move together obviously, so if the extensions given by an equation then the speed will, be given by the rate of change foe xtension