Differential Equations

Question 1

Write the definition of total differentiability.

Answer 1

Check in book.

Question 2

Totally differentiability implies what?

Continuously differentiable implies what?

Answer 2

Continuity

Locally Lipschitz-continuous

Question 3

How do you show that there exists a unique solution?

Answer 3

By the Picard-Lindelof existence and uniqueness theorem which says: let the function be continuous and locally-Lipschitz then there exists a unique solution.
(Show it is global then it is locally)

Question 4

What are the maximal solutions?

Answer 4

t_ =-infinity (i)
Take limit t -> t+
Unbounded (ii)
Solution approaches the boundary (ii)

Visa versa for t+=infinity

Question 5

If an ODE is dependent on t, what does this mean for its solution?

Answer 5

A solution exists which is unique

Question 6

Write the formula for the Picard-Lindelof iteration.

Answer 6

Check in book.

Question 7

What is Gronwall’s lemma?

Answer 7

Set z(t)=||x(t)-y(t)||^2
Differentiate 
And put into matrix form

Question 8

How do you calculate the fundamental matrix solution?

Answer 8

Find the eigenvalues then the corresponding eigenvectors and it will be: 
C1exp(integral of a1 dt)(v1)
Write X(t) in column vectors and at t0 the matrix should be the identity.

Question 9

For transformation of higher orders what is the new function we define?

Answer 9

Check on first page of book.

Question 10

What is the variation of constants formula?

Answer 10

Check in book

Question 11

What are the different methods of integration?

Answer 11

Check on formula sheet

Question 12

What are the steps to calculate partial differential equations?

Answer 12

1. Write the initial conditions
2. Dx/Dsigma=a and Dt/Dsigma=b
3. Find x and t by integration
4. Then rearrange in terms of sigma and eta.
5. Du/Dsigma=c find u by integration
6. Then substitute so it is terms of x and t

Question 13

What are the steps for the reduction of order?

Answer 13

1. We start with a solution x(t) and the second solution is y(t)=x(t)v(t)
2. Find y’ and y’’ and substitute into original equation the v term should drop.
3. Use change of variables v’=w and v’‘=w’
4. Solve using method of integration to obtain w(t)
5. Recalling change of variables v(t)=integral of w(t)
6. Thus y(t)=v(t)*x(t) is the second solution

Question 14

For repeated roots what do we do?

Answer 14

Substitute lambda and find v1 and then find the kernel of (A-lambdaI)^2 to obtain v2. Then the solution x(t)=c1e^lambdat(v1) + c2e^lambda*t(v2 + t(v1))

Question 15

For complex root what do we do?

Answer 15

Find v1 and the form e^lambda*it into cos and sine. Multiply into the matrix and split into real and imaginary parts. Then x(t) is c1u + c2v

Question 16

How to do method of undetermined coefficients

Answer 16

1. Write characteristic equation and find eigenvalues and form complementary solution: c1e^lambda1t+c3e^lambda2t
2. Find a general particular solution. Differentiate appropriately and substitute to find coefficients.
3. Then x(t)=xc+xp

Question 17

What is the existence and uniqueness theorem?

Answer 17

Let f be continuous,
Assume in addition that f is locally Lipschitz continuous wrt x in G
Then for every (t0,x0) in G there exists an epsilon bigger than 0 such that the solution of the IVP exists and is unique in the interval I=(t0-3,t0+3)

Question 18

What is the definition for Gronwall’s lemma?

Answer 18

Let I=[t0,t1) where t1= infinity is allowed. Let z and alpha be two continuous functions. Let z be differentiable on I and satisfy z’(t)