4.2 Direction fields and numerical methods  (Page 6/14)

$y\prime =y^2-1$

$y\prime =y-x$

$y\prime =1-y^2-x^2$

$y\prime =t^2\text{sin}y$

$y\prime =3y+xy$

$y\prime =\mathrm{-3}y$

$y\prime =\mathrm{-3}t$

$y\prime =e^t$

$y\prime =\frac{1}{2}y+t$

$y\prime =\text{−}ty$

$y\prime =t\text{sin}y$

$y\prime =\text{−}t\text{cos}y$

$y\prime =t\text{tan}y$

$y\prime ={\text{sin}}^{2}y$

$y\prime =y^2{t}^3$

$y\prime =\mathrm{-3}y,y\left(0\right)=1$

$y\prime =t^2$

$y^{\prime }=3t-y,y(0)=1.$ Exact solution is $y=3t+4e^{\text{−}t}-3$

$y^{\prime }=y+t^2,y(0)=3.$ Exact solution is $y=5e^t-2-t^2-2t$

$y^{\prime }=2t,y(0)=0$

[T] $y\prime =e^{(x+y)},y(0)=\mathrm{-1}.$ Exact solution is $y=\text{−}\text{ln}(e+1-e^x)$

$y^{\prime }=y^2\text{ln}(x+1),y(0)=1.$ Exact solution is $y=-\frac{1}{(x+1)(\text{ln}(x+1)-1)}$

$y^{\prime }=2^x,y(0)=0,$ Exact solution is $y=\frac{2^x-1}{\text{ln}(2)}$

$y^{\prime }=y,y(0)=\mathrm{-1}.$ Exact solution is $y=\text{−}{e}^x.$

$y^{\prime }=\mathrm{-5}t,y(0)=\mathrm{-2}.$ Exact solution is $y=-\frac{5}{2}{t}^2-2$

Show that, by our assumption that the total population size is constant $(S+I=N),$ you can reduce the system to a single differential equation in $I\text{:}I\prime =c(N-I)I-rI.$

Assuming the parameters are $c=0.5,N=5,$ and $r=0.5,$ draw the resulting directional field.

[T] Use computational software or a calculator to compute the solution to the initial-value problem $y\prime =ty,y\left(0\right)=2$ using Euler’s Method with the given step size $h.$ Find the solution at $t=1.$ For a hint, here is “pseudo-code” for how to write a computer program to perform Euler’s Method for $y\prime =f(t,y),y(0)=2\text{:}$

Create function $f(t,y)$

Define parameters $y\left(1\right)=y_0,t\left(0\right)=0,$ step size $h,$ and total number of steps, $N$

Write a for loop:

for $\text{k}=1\text{to N}$

$\text{fn}=\text{f}\left(\text{t}\left(\text{k}\right),\text{y}\left(\text{k}\right)\right)$

$\text{y}\left(\text{k+1}\right)=\text{y}\left(\text{k}\right)+\text{h*fn}$

$\text{t}\left(\text{k+1}\right)=\text{t}\left(\text{k}\right)+\text{h}$

Solve the initial-value problem for the exact solution.

Draw the directional field

$h=1$

[T] $h=10$

[T] $h=100$

[T] $h=1000$

[T] Evaluate the exact solution at $t=1.$ Make a table of errors for the relative error between the Euler’s method solution and the exact solution. How much does the error change? Can you explain?

Show that $y=2e^{\mathrm{-2}x}$ solves this initial-value problem.

Draw the directional field of this differential equation.

[T] By hand or by calculator or computer, approximate the solution using Euler’s Method at $t=10$ using $h=5.$

[T] By calculator or computer, approximate the solution using Euler’s Method at $t=10$ using $h=100.$

[T] Plot exact answer and each Euler approximation (for $h=5$ and $h=100)$ at each $h$ on the directional field. What do you notice?