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(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

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  • $\begingroup$ This equation is linear; you might want to adjust the title. Regards. $\endgroup$ – Robert Lewis Jan 25 '15 at 6:55
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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.

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