Write the definition of total differentiability.
Check in book.
Totally differentiability implies what?
Continuously differentiable implies what?
Continuity
Locally Lipschitz-continuous
How do you show that there exists a unique solution?
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)
What are the maximal solutions?
t_ =-infinity (i)
Take limit t -> t+
Unbounded (ii)
Solution approaches the boundary (ii)
Visa versa for t+=infinity
If an ODE is dependent on t, what does this mean for its solution?
A solution exists which is unique
Write the formula for the Picard-Lindelof iteration.
Check in book.
What is Gronwall’s lemma?
Set z(t)=||x(t)-y(t)||^2 Differentiate And put into matrix form
How do you calculate the fundamental matrix solution?
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.
For transformation of higher orders what is the new function we define?
Check on first page of book.
What is the variation of constants formula?
Check in book
What are the different methods of integration?
Check on formula sheet
What are the steps to calculate partial differential equations?
- Write the initial conditions
- Dx/Dsigma=a and Dt/Dsigma=b
- Find x and t by integration
- Then rearrange in terms of sigma and eta.
- Du/Dsigma=c find u by integration
- Then substitute so it is terms of x and t
What are the steps for the reduction of order?
- We start with a solution x(t) and the second solution is y(t)=x(t)v(t)
- Find y’ and y’’ and substitute into original equation the v term should drop.
- Use change of variables v’=w and v’‘=w’
- Solve using method of integration to obtain w(t)
- Recalling change of variables v(t)=integral of w(t)
- Thus y(t)=v(t)*x(t) is the second solution
For repeated roots what do we do?
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))
For complex root what do we do?
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
How to do method of undetermined coefficients
- Write characteristic equation and find eigenvalues and form complementary solution: c1e^lambda1t+c3e^lambda2t
- Find a general particular solution. Differentiate appropriately and substitute to find coefficients.
- Then x(t)=xc+xp
What is the existence and uniqueness theorem?
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)
What is the definition for Gronwall’s lemma?
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)
