Exercises

(1)
How many asymptotically unstable equilibrium solutions does \(y^{\,\prime }=y^5-y^3\) have?
(2)
Consider the following statements about the constant function \(\displaystyle y=e^2\) in regards to the first order autonomous differential equation \(\displaystyle y^{\,\prime }=\frac {2-\ln {(y)}}{1+y^2}\). Determine which of the following statements is true.
(a)
It is not an equilibrium solution of this equation.
(b)
It is an asymptotically stable equilibrium solution of this equation.
(c)
It is an asymptotically unstable equilibrium solution of this equation.
(d)
It is an asymptotically semistable equilibrium solution of this equation.
(e)
None of the above are true.
(3)
Which could be the limit as \(t\) goes to infinity of a non-constant solution to \(y^{\,\prime }=y^4-3y^3+2y^2\)? Select all that apply.
(a)
\(-\infty \)
(b)
\(-2\)
(c)
\(-1\)
(d)
\(0\)
(e)
\(1\)
(f)
\(2\)
(g)
\(\infty \)
(4)
Let \(0<a<5\). Suppose we are given that the solution \(y(t)\) of the IVP \(y^{\,\prime }=(e^y - 3)(y^2-ay)\), \(y(0)=2\) satisfies \(\displaystyle \lim _{t\rightarrow \infty }y(t)=\ln {(3)}\). Find a possible value of \(a\).
(5)
A second order autonomous equation is a differential equation of the form \(y^{\,\prime \prime }=f(y,y^{\,\prime })\). In the special case where \(y^{\,\prime \prime }=f(y)\), we can multiply both sides by \(y^{\,\prime }\) to get \(y^{\,\prime }y^{\,\prime \prime }=f(y)y^{\,\prime }\), which we can integrate with respect to the independent variable (using that \(y^{\,\prime \prime } \mathrm {d}t=\mathrm {d}(y^{\,\prime })\) and \(y^{\,\prime } \mathrm {d}t=\mathrm {d}y\)) to get a separable differential equation that can be solved.

Let \(y(t)\) be the solution to the second order IVP
\(y^{\,\prime \prime }=-\sin (y)\), \(y(0)=0\), \(y^{\,\prime }(0)=2\). Determine \(t\) such that \(y(t)=1\). Round to the nearest hundredth.

Hint: you may find \(\cos ^2(\theta )=\dfrac {1}{2}\left (1+\cos (2\theta )\right )\) and
\(\displaystyle \int \sec (\theta )\,\mathrm {d}\theta =\ln |\sec (\theta )+\tan (\theta )|+C\) useful.