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Exercises relating to fundamental properties of ODEs.

Which of the curves below is a solution of the ODE illustrated by the slope field?

A B C D E
Which of the curves below is a solution of the ODE illustrated by the slope field?

A B C D E
Trace the solution of the given ODE which begins at the point A. Which of the other labelled points will the solution pass through?

B C D E
The ODE $y' = 2y (1-y)$ has one asymptotically stable constant solution. It is
Trace the solution of the given ODE which begins at the point A. Which of the other labelled points will the solution pass through?

B C D E
The ODE $y' = x^3 - 3x y^2$ has noonetwothree constant solutions.
Let $y(x)$ be the solution to the initial value problem $y' = y$ with $y(0) = 1$. Use Euler’s Method with step size $h = 1$ to approximate the value of $y(4)$. Fill in your work in the table below.

 $x$ $y$ $y + y h$ $0$ $1$ $2$ $1$ $\answer {2}$ $\answer {4}$ $\answer {2}$ $\answer {4}$ $\answer {8}$ $\answer {3}$ $\answer {8}$ $\answer {16}$ $\answer {4}$ $\answer {16}$

Let $y(x)$ be the solution to the initial value problem $y' = x-y$ with $y(0) = 1$. Use Euler’s Method with step size $h = 1$ to approximate the value of $y(5)$. Fill in your work in the table below.

 $x$ $y$ $(x - y)$ $y + (x - y) h$ $\answer {0}$ $\answer {1}$ $\answer {-1}$ $\answer {0}$ $\answer {1}$ $\answer {0}$ $\answer {1}$ $\answer {1}$ $\answer {2}$ $\answer {1}$ $\answer {1}$ $\answer {2}$ $\answer {3}$ $\answer {2}$ $\answer {1}$ $\answer {3}$ $\answer {4}$ $\answer {3}$ $\answer {1}$ $\answer {4}$ $\answer {5}$ $\answer {4}$

Is the function $y = x-1$ a solution of the ODE $y' = x-y$?

Yes No
If yes, is it a solution of the IVP $y' = x-y$ and $y(0) = 1$?
Yes No Not applicable
Is the function $y = x -1 + 2 e^{-x}$ a solution of the IVP $y' = x-y$ and $y(0) = 1$?
Yes No
Given this, what is the difference between the approximated value of $y(5)$ that you calculated and the exact value. If you get a negative answer, take its absolute value: