Lesson test: Existence, uniqueness and stepping forward

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Existence, uniqueness and stepping forward

1.[2p]
In Picard's theorem, what does the condition on $\partial f / \partial y$ provide?

Uniqueness, since continuity of $f$ alone gives existence but permits several solutions

Correct

Existence, since without it no solution need exist

The answer is: Uniqueness, since continuity of $f$ alone gives existence but permits several solutions

A guarantee that the solution exists for all time

The answer is: Uniqueness, since continuity of $f$ alone gives existence but permits several solutions

A bound on the error of Euler's method

The answer is: Uniqueness, since continuity of $f$ alone gives existence but permits several solutions

2.[1p]
If $f (t, y)$ is continuous near the initial point, the initial value problem has exactly one solution there.

True

The answer is: False

False

Correct

3.[3p]
For $y' = y^{2}$ with $y (0) = 2.5$ , at what time does the solution cease to exist?

CorrectNot quite: 0.4

4.[2p]
Why does $y' = \sqrt{y}$ with $y (0) = 0$ have infinitely many solutions?

$\partial f / \partial y$ is unbounded at $y = 0$ , so the Lipschitz condition fails there

Correct

The equation is not separable

The answer is: $\partial f / \partial y$ is unbounded at $y = 0$ , so the Lipschitz condition fails there

The square root is not defined for negative $y$

The answer is: $\partial f / \partial y$ is unbounded at $y = 0$ , so the Lipschitz condition fails there

The initial value is zero, which no solution can pass through

The answer is: $\partial f / \partial y$ is unbounded at $y = 0$ , so the Lipschitz condition fails there

5.[3p]
Apply Euler's method to $y' = t + y$ with $y (0) = 1$ and step $h = 0.2$ . What is the estimate after two steps, at $t = 0.4$ ?

CorrectNot quite: 1.48

6.[2p]
Euler's method gives an error of 0.28 at step size $h = 0.25$ on a certain problem. What error does its first order accuracy predict at $h = 0.0625$ ?

CorrectNot quite: 0.07

7.[2p]
Put these methods in order of increasing accuracy, from the largest error at a given step size to the smallest.

Improved Euler (Heun)
Classical fourth order Runge-Kutta
Euler

Show the answer

c, a, b

8.[3p]
Which statements about numerical solutions are correct?

Select all that apply

A solver keeps printing values after the true solution has blown up

Correct

Runge-Kutta of fourth order costs four evaluations of $f$ per step

Correct

A stiff equation can force a tiny step for stability even where accuracy would allow a large one

Correct

Reducing the step size always reduces the total error, however far it is reduced

The answer is: A solver keeps printing values after the true solution has blown up, Runge-Kutta of fourth order costs four evaluations of $f$ per step, A stiff equation can force a tiny step for stability even where accuracy would allow a large one

9.[2p]
Why does a first order linear equation need no local hedge in its existence statement?

Its right side has $y$ -derivative $- p (t)$ , so the hypotheses hold wherever the coefficients are continuous

Correct

Because linear equations are always separable

The answer is: Its right side has $y$ -derivative $- p (t)$ , so the hypotheses hold wherever the coefficients are continuous

Because its solutions are always bounded

The answer is: Its right side has $y$ -derivative $- p (t)$ , so the hypotheses hold wherever the coefficients are continuous

Because the integrating factor is never zero

The answer is: Its right side has $y$ -derivative $- p (t)$ , so the hypotheses hold wherever the coefficients are continuous

1.[2p] In Picard's theorem, what does the condition on ∂f/∂y provide?

2.[1p] If f(t,y) is continuous near the initial point, the initial value problem has exactly one solution there.

3.[3p] For y′=y2 with y(0)=2.5, at what time does the solution cease to exist?

4.[2p] Why does y′=y with y(0)=0 have infinitely many solutions?

5.[3p] Apply Euler's method to y′=t+y with y(0)=1 and step h=0.2. What is the estimate after two steps, at t=0.4?

6.[2p] Euler's method gives an error of 0.28 at step size h=0.25 on a certain problem. What error does its first order accuracy predict at h=0.0625?

7.[2p] Put these methods in order of increasing accuracy, from the largest error at a given step size to the smallest.

8.[3p] Which statements about numerical solutions are correct?

9.[2p] Why does a first order linear equation need no local hedge in its existence statement?

1.[2p]
In Picard's theorem, what does the condition on $\partial f / \partial y$ provide?

2.[1p]
If $f (t, y)$ is continuous near the initial point, the initial value problem has exactly one solution there.

3.[3p]
For $y' = y^{2}$ with $y (0) = 2.5$ , at what time does the solution cease to exist?

4.[2p]
Why does $y' = \sqrt{y}$ with $y (0) = 0$ have infinitely many solutions?

5.[3p]
Apply Euler's method to $y' = t + y$ with $y (0) = 1$ and step $h = 0.2$ . What is the estimate after two steps, at $t = 0.4$ ?

6.[2p]
Euler's method gives an error of 0.28 at step size $h = 0.25$ on a certain problem. What error does its first order accuracy predict at $h = 0.0625$ ?

7.[2p]
Put these methods in order of increasing accuracy, from the largest error at a given step size to the smallest.

8.[3p]
Which statements about numerical solutions are correct?

9.[2p]
Why does a first order linear equation need no local hedge in its existence statement?