Differential Equations I

Question 1

Differential Equations I

ODE (D)

Answer 1

An equation of the form G(x,y,y’,y’‘,…,y^(n))=0

Question 2

Differential Equations I

First-order ODE with data (D)

Answer 2

y’(c)=f(x,y(x)) with y(a)=b

Question 3

Differential Equations I

A function f(x,y) on a rectangle R is Lipschitz in y if (D)

Answer 3

There exists L st

¦ f(x,u) - f(x,v) ¦ <= L¦u-v¦ for all (x,u) in R all (x,v) in R

Question 4

Differential Equations I
What is the integral equation of the IVP
y’(x) = f(x, y(x) ) with y(a)=b (Q)

Answer 4

y(x) = b + [a]int[x] f( t, (y(t) ) dt

Question 5

Differential Equations I

What is the iteration process and define the error terms used in the 1st proof of Picard’s theroem (Q)

Answer 5

y0(x) = b
yn+1(x) = b + [a]int[x] f( t, y(t) ) dt

e0(x)=b
en(x) = yn(x) - yn-1(x)

Question 6

Differential Equations I
Write yn(x) in terms of the error terms in the 1st proof of Picard's Theorem (Q)

Answer 6

yn(x) = [0]sum[n] ek(x)

Question 7

Differential Equations I

State Picard’s theorem in full (T)

Answer 7

Let f : R -> Reals be a function on R which satifies both of
P(i) a) f is continuous on R and so bounded, say ¦f(x,y)¦=M. b) Mh<=k
P(ii) f is Lipschitz on R so ¦f(x,u)-f(x,v)¦<=L¦u-v¦ for (x,u), (x,v) in R

Then the IVP y’(x) = f(x,y) with y(a)=b has a unique solution in R

( R = { (x,y) : ¦x-a¦<=h, ¦y-b¦<=k )

Question 8

Differential Equations I

State the 4 claims used in the 1st proof of Picard’s theorem (T)

Answer 8

Claim 1: Each yn is well-defined and continuous and ¦yn-b¦<=k for x in a-h, a+h]

Claim 2: For n>=1 and ¦x-a¦<=h.
en(x)<= L^n-1 * M * ¦x-a¦^n / n!

Claim 3: The yn= [1]sum[n] ej(x) converge uniformly to a continuous function y∞ and this is a solution of the IVP

Claim 4: The solution is unique among all functions
y: [a-h, a+h] -> [b-k, b+k]

Question 9

Differential Equations I

Give the rough proof idea for the 4 claims in Picard’s theorem (T)

Answer 9

Claim 1: a) law of integration, b) induction using M and mh<=k

Claim 2: Induction using triangle and Lipschitz. First show en(x) <= L [a]int[x] en-1(x) dt

Claim 3: Use claim 2’s bound and the Weierstrass M-test. To show sol, take limit in integral equation and swap limit and integral

Claim 4: Assume 2 the same. Show difference = 0 using induction similarly to the e(n) inequality

Question 10

Differential Equations I

1st Picard proof extension: Conditions for extending Picard’s proof to y in reals, x in [a-h, a+h] (Q)

Answer 10

f(x,y) is continuous for all y for all x in [a-h, a+h].

f satisfies a global Lipschitz condition. That is, for all y, for all x in [a-h, a+h]

Question 11

Differential Equations I

1st Picard proof extension: condition for extending Picard’s theorem to all the reals (Q)

Answer 11

There exists a global Lipschitz condition on [a-h, a+h] for all h>0

Question 12

Differential Equations I

State Gronwall’s Inequality (T)

Answer 12

Let A>=0 and b>=0. Let v be non negative and let

v(x)<= b + A¦ [a]int[x] v(s) ds ¦
Then v(x) <= b * e^A¦x-a¦

Question 13

Differential Equations I

Give the proof idea for Gronwall’s Inequality (T)

Answer 13

Look at x>=a Take V(x) = [a]int[x] v(s) ds.
Then V’(x)=v(x) and V’(x)<=b+AV(x).
Solve this ODE
Similarly for x<=a

Question 14

Differential Equations I

How can you use Gronwall’s to show continuous dependance and uniqueness? (Q)

Answer 14

Take y and z both solutions where y(a)=b z(a)=c. Then bound ¦y-z¦ above by ¦b-c using Gronwall

Question 15

Differential Equations I

Define a contraction (D)

Answer 15

We say a map T: Ch,k -> Ch,k is a contraction if there exists a 0

Question 16

Differential Equations I

State the contraction mapping theorem (T)

Answer 16

Let X be a complete metric space and let T: X->X be a contraction. Then there is a unique fixed point in X ie. Ty=y

Question 17

Differential Equations I

State Picard’s theorem via CMT (T)

Answer 17

Let f: R-> Reals be a function defined on the usual rectangle R. Let f satisfy P(i) and P(ii) and let n>0 be so that Ln<1 and Mn<=k.
Then the IVP has a unique solution in [a-n, a+n]

Question 18

Differential Equations I

Picard’s proof via CMT strategy (T)

Answer 18

y sol of IVP <=> y sol of integral equation.

Let (Ty)(x) = b + [a]int[x] f( s, y(s) )) ds. Then y is a fixed point

Claim 1: If Mn<=k then T:Cn,k->Cn,k. (Show ¦¦Ty-b¦¦sup<=k)
Claim 2: If Ln<=1 then T is a contraction with K=Ln. (Show ¦¦Ty-Tz ¦¦sup<=K¦¦y-z¦¦sup

Then choose n < what it needs to be

Question 19

Differential Equations I

Picards via CMT: Conditions to extend to [a-h, a+h] (Q)

Answer 19

Mh<=k

Question 20

Differential Equations I

Picard’s via CMT: Extending to [a-h,a+h] proof idea? (Q)

Answer 20

Look in n-length segments for solutions with initial data being that the boundaries are the same. Do this until you reach h. For rach solution RTP ¦¦Ty-b¦¦sup<=k and length of segment <1/L

Question 21

Differential Equations I

Picard’s via CMT: Conditions for global solution (Q)

Answer 21

f continuous for all x in [a-h, a+h] and all y. global Lipschitz condition on [a-h,a+h]xReals

Question 22

Differential Equations I

Picard’s via CMT: What do we mean by global solution in this case (Q)

Answer 22

A unique solution on all of [a-h, a+h] regardless of y

Question 23

Differential Equations I

1st Picard proof extension: Extending Picard’s proof to y in reals, x in [a-h, a+h] proof strategy (T)

Answer 23

In claim 1 we don’t need ¦yn-b¦<=k. So don’t need Mh<=k

When does M appear again? In claim 2. But here we only need M=sup¦f(x,b)¦ which exists as continuous and bounded.

Question 24

Differential Equations I

Picard’s via CMT: Global solution proof idea (T)

Answer 24

Claim 1 doesn’t need Mh<=k and we obtain a sol on [a-n,a+n] for n<1/L using Picard CMT. Then use extension method. This proof is easier as we don’t need ¦¦Ty-b¦¦<=k. We obtain a sol on all of [a-h,a+h]

Question 25

Differential Equations I

Comparison of 2 Picard methods (Q)

Answer 25

CMT is easier and quicker as most work is already done. CMT also gives uniqueness for free.

The methods are essentially equivalent as convergence in sup norm <=> uniform convergence

CMt produces less delicate result as the range for x is smaller

Question 26

Differential Equations I

Pair of 1st order ODEs (vector form too) (D)

Answer 26

y’1(x)=f1(x, y1(x), y2(x) )
y’2(x)=f2(x, y1(x), y2(x) )
y1(a)=b1, y2(a)=b2

in Vector form:
Y’(x)=F(x,Y(x))
Y(a)=B

Question 27

Differential Equations I

State Picard’s theorem for systems (T)

Answer 27

Let f1, f2 : S -> Reals be functions which satisfy H(i) and H(ii). S = [a-h,a+h] x Bk(b). Then there exists 0

Question 28

Differential Equations I

Picard’s theorem for systems proof idea (T)

Answer 28

Use metric space Cn=C([a-h,a+h];Bk(b)) with metric
¦¦Y¦¦sup = sup¦¦Y(x)¦¦1 := sup ( ¦y1¦+¦y2¦ ). Then again define T, prove its in the right space and prove its a contraction.

Question 29

Differential Equations I

Picard for systems: Conditions to extend to [a-h,a+h] (not all y) (Q)

Answer 29

Mh<=k

Question 30

Differential Equations I

Picard for systems: Condition for global solution (Q)

Answer 30

Global Lipschitz

Question 31

Differential Equations I

Picard for 2nd order: What is the 2nd order form and how to convert it into a system? (Q)

Answer 31

y’‘+p(x)y’+q(x)y=r(x) with y(a)=b, y’(a)=c

Let z=y’ so
y’=z:=f1(x,y,z)
z’=-pz-qy-r:=f2(x,y,z)
y(a)=b, z(a)=c