Problem 58
Solve the following initial-value problems starting from \(y(t=0)=1\) and \(y(t=0)=-1 .\) Draw both solutions on the same graph\(\frac{d y}{d t}=2\)
Problem 59
Solve the following initial-value problems starting from \(y_{0}=10 .\) At what time does \(y\) increase to 100 or drop to \(1 ?$$\frac{d y}{d t}=4 t\)
Problem 59
Estimate the following solutions using Euler's method with \(n=5\) steps over the interval \(t=[0,1] .\) If you are able to solve the initialvalue problem exactly, compare your solution with the exact solution. If you are unable to solve the initial-value problem, the exact solution will be provided for you to compare with Euler's method. How accurate is Euler's method? $$ y^{\prime}=-5 t, y(0)=-2 . \text { Exact solution is } y=-\frac{5}{2} t^{2}-2 $$
Problem 60
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 \(I: I^{\prime}=c(N-I) I-r I\)
Problem 60
Solve the following initial-value problems starting from \(y_{0}=10 .\) At what time does \(y\) increase to 100 or drop to \(1 ?$$\frac{d y}{d t}=4 y\)
Problem 61
Solve the following initial-value problems starting from \(y_{0}=10 .\) At what time does \(y\) increase to 100 or drop to \(1 ?$$\frac{d y}{d t}=-2 y\)
Problem 61
Solve the generic equation \(y^{\prime}=a x+y .\) How does varying \(a\) change the behavior?
Problem 62
Solve the following initial-value problems starting from \(y_{0}=10 .\) At what time does \(y\) increase to 100 or drop to \(1 ?$$\frac{d y}{d t}=-2 y\)
Problem 62
Solve the generic equation \(y^{\prime}=a x+x y .\) How does varying \(a\) change the behavior?
Problem 63
Solve the following initial-value problems starting from \(y_{0}=10 .\) At what time does \(y\) increase to 100 or drop to \(1 ?$$\frac{d y}{d t}=e^{4 t}\)
