(a) Draw a direction field for the given differential equation. How do solutions appear to
behave as $t → 0$? Does the behavior depend on the choice of the initial value $a$? Let $a_{0}$
be the value of $a$ for which the transition from one type of behavior to another occurs.
Estimate the value of $a_{0}$.
(b) Solve the initial value problem and find the critical value $a_{0}$exactly.
(c) Describe the behavior of the solution corresponding to the initial value $a_{0}$.
$$ty'+(t+1)y=2te^{-t}, y(1)=a$$
What i tried,
I started out by solving this equation using the method of integrating factors. My integreating factor is $u(t)=te^{t}$.I then multiplying both sides of the eqn by the integrating factor to get, $$yte^{t}=\int{2t}dt$$ and solving it to get $$y=\frac{t^{2}+c }{te^{t}} $$ for the general solution. Then i plug in the initial conditions to solve the ODE. While solving the ODE is not a problem for me, what im stuck is at the remaining portions of the question, specifically im unsure how the behavior depend on the choice of the initial value $a$ and how the behavior of the solution corresponding to the initial value $a_{0}$ as well a plotting the direction field for this ODE. What i believe is that the transition point behaviour have something to do with the initial conditions $y(1)=a$ but im not too sure about that. COuld anyone please explain. Thanks
Best Answer
For part a), the problem asks you to draw the direction field without solving the ODE. Writing it in this form:
$$y'=2e^{-t}-(1+\frac{1}{t})y$$
Try some $y$ values. For example, let $y=0,y'=2e^{-t}$. That will give you the direction field on $t$ axis. They are arrows pointing upward, but approaching 0 direction, toward some equilibrium value.
Let $y=1, y'=2e^{-t}-(1+\frac{1}{t})$. This gives you the direction field on the line $y=1$. They are arrows pointing downward, approaching $0$ direction, toward the same equilibrium value.
Especially you should look at the direction of the points when $t=1$.
So what is the critical value of $y$ at $t=1$, such that the field changed direction? I believe you can find that.