Thread: Question on autonomous differential equation

I have the differential equation:

dx/dt=x²+5x-6
I have to find the equilibria and solve them.

Second I have to answer the question:
Let x₁(t) be its particular solution such that x₁(t₁)=-7 - for some time instant t1. Is it possible that x₁(t)=2 for some t? Explain.

To be honest, I have no clue about how to answer to the second question...

I find the points by setting:
0=x²+5x-6
x=-6 or x=1

to get the equilibrium points:
df/dx=2x+5
= 17 for x=-6 and 7 for x=1

so both points are unstable.
Is this true? - How can I answer the second question.

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Another quick question:
Is the differential equation linear? Is it separable?
x'=x²+e^t+e^x

No it's not linear because it contains the second power of x. It's not separable, because it can not be written as dx/h(x)=t*g(t)
Is this the right answer?

For x=-6 you get df/dx = -7, so x=-6 is stable point.

So I guess but I cannot recall (took ODE courses 3-4 years ago, a bit rusty) that if at some time our solution equals -7, then because x=-6 is a stable solution all solutions will be attracted to it, thus there cannot be a time t such that x_1(t)=2.