# Thread: ODE Autonomous Help

1. ## ODE Autonomous Help

So I have a problem that, with help, I figured out the solution to part A&B. I overlooked a part C.

So the differential equation is (dy/dx) = (y+4)(y-5)

And the question is:

Solve the differential equation in problem 1 above for y(x) with y(0) = 1, and find lim y(x) as x approaches infinity. Would you just distribute (y+4)(y-5) and integrate, then plug in the initial condition? I drew a slope field with solutions curves for general solutions in Part B, and they also ask does this limit agree with the pictures in question 1? It is a 1 which I found to be approaching a horizontal asymptote. I'm guessing thats what they are asking to confirm?

2. ## Re: ODE Autonomous Help

No, the equation is separable.

\displaystyle \begin{align*} \frac{\mathrm{d}y}{\mathrm{d}x} &= \left( y + 4 \right) \left( y - 5 \right) \\ \frac{1}{\left( y + 4 \right) \left( y - 5 \right) }\,\frac{\mathrm{d}y}{\mathrm{d}x} &= 1 \\ \int{ \frac{1}{\left( y + 4 \right) \left( y - 5 \right) } \,\frac{\mathrm{d}y}{\mathrm{d}x}\,\mathrm{d}x} &= \int{ 1\,\mathrm{d}x} \\ \int{ \frac{1}{\left( y + 4 \right) \left( y - 5 \right) } \,\mathrm{d}y} &= \int{ 1\,\mathrm{d}x} \end{align*}

Go from here. You can evaluate the left hand integral using Partial Fractions.

3. ## Re: ODE Autonomous Help

Actually, you can answer this question without explicitly solving the differential equation. First, since dy/dx= (y+5)(y-4)= 0 for y= -5 and y= 4, y(x)= -5 for all x, and y(x)= 4x, for all x are "constant" solutions. Further, y= 1 lies between -5 and 4, dy/dx= (1+ 5)(1- 4)= 6(-3)= -18 so the y is decreasing at that point, and only can change sign at places where dy/dx= 0, that is, when y= -5 and 4. Finally, two solution curves cannot cross so the function, y(x), must decrease down to -5 as x goes to infinity.

4. ## Re: ODE Autonomous Help Originally Posted by HallsofIvy Actually, you can answer this question without explicitly solving the differential equation.
Except the question says to solve the DE...

5. ## Re: ODE Autonomous Help

You are right. What I said was probably relevant to parts A and B! Also, ProveIt, the equations in your 'signature' don't show up properly. You might want to change the [tex] in it.