Section 6.2 Solving Initial Value Problems
Exercises Exercises
Supplemental HW Problems.
ODE = Ordinary Differential Equation
IVP = Initial Value Problem
1.
Consider the Differential Equation
\begin{equation*}
y’ = y-4e-t
\end{equation*}
with the IVP \(y(0) = 1.\)
Using a CAS, graph the slope field for the ODE, and highlight the IVP solution curve.
Using a technique from chapter 2, solve the IVP above.
Now using a Laplace Transform technique, solve the IVP above.
Which do you prefer and why?
2.
Now, consider the following Spring-Mass system modeled with the 2nd order Linear ODE:
\begin{equation*}
x'' + 4x = \sin(3t), x(0) = x’(0) = 0
\end{equation*}
Unpack this equation; describe in words what is going on here, and use a CAS to graph \(x(t).\)
Use a non-Laplace argument to solve the IVP for \(x(t).\)
Use a Laplace Transform to solve the IVP for \(x(t).\)
Which do you prefer and why?
3.
A spring-mass system is modeled with the 2nd order Liner ODE:
\begin{equation*}
x'' + 5x’ + 6x = 0; x(0) = 5; x’(0) = 0
\end{equation*}
Unpack that equation – describe in words the forces being summed here, and what the initial condition represents. Without solving or graphing, what do you expect this system to do, long-term?
Use a CAS to graph \(x(t).\) What is the long-term behavior, from your graph?
Solve the system. Use either a Laplace argument, or a characteristic polynomial argument. If we let \(t\) go to infinity, what happens?
4.
To the above spring-mass system, we attach a mechanism that adds the forcing function \(f(t) = \cos(t),\) so our new ODE is:
\begin{equation*}
x'' + 5x’ + 6x = \cos(t); x(0) = 5; x’(0) = 0
\end{equation*}
Unpack that equation – describe in words the forces being summed here, and what the initial condition represents. Without solving or graphing, what do you expect this system to do, long-term?
Use a CAS to graph \(x(t).\) What is the long-term behavior, from your graph?
Solve the system. Use either a Laplace argument, or a characteristic polynomial argument. If we let \(t\) go to infinity, what happens?
5.
We change the situation above – the forcing function is now intermittent, and only acts on the system over the interval \(t \geq 5.\)
Thinking about our spring-mass system with this force added, do we expect the short-run behavior to be different? What about the long-term behavior?
Solve this system without a Laplace method. We’ll need to have two different solutions – before and after the force acts – and different initial conditions for each.
Solve this system using a Laplace method.
Does it matter if the endpoint is included? Thinking about the system, what changes would you expect if the force acted over the interval \(t \gt 5\) instead?
6.
One more change – same spring-mass system we’ve been looking at,
\begin{equation*}
x'' + 5x’ + 6x = f(t), x(0) = 5, x’(0) = 0.
\end{equation*}
Now, however, we apply the force \(f(t) = \cos(t)\) over the interval \(5 \lt t \lt 10.\)
Thinking about our spring-mass system with this force added, what do we expect our long-term behavior to be?
Solve this system without a Laplace method. We’ll need to have three different solutions – before, during, and after the force acts – and different initial conditions for each.
Solve this system using a Laplace method.
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