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

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

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

Trace the solution of the given ODE which begins at the point A. Which of the other labelled points will the solution pass through? The ODE $y' = x^3 - 3x y^2$ has 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.

$$y(4) \approx \answer {16}.$$

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.

$$y(5) \approx \answer {4}.$$ Is the function $y = x-1$ a solution of the ODE $y' = x-y$?

If yes, is it a solution of the IVP $y' = x-y$ and $y(0) = 1$? Is the function $y = x -1 + 2 e^{-x}$ a solution of the IVP $y' = x-y$ and $y(0) = 1$? 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: $$| y_{\text {approx}}(5) - y_{\text {exact}}(5)| = \answer {2 e^{-5}} \approx 0.0135.$$