Exercises

(1)
Determine which first order differential equation has this slope field.
A direction field. [Picture]
Figure 1: A direction field (also known as a slope field)
(a)
\(y^{\,\prime }=xy\)
(b)
\(y^{\,\prime }=x^2y\)
(c)
\(y^{\,\prime }=xy^2\)
(d)
\(y^{\,\prime }=x^2y^2\)
(e)
\(y^{\,\prime }=x/y\)
(f)
\(y^{\,\prime }=y/x\)
(2)
Find a real value of the parameter \(k\) such that \(y=e^{kt}\) is a solution to \(y^{\,\prime \prime }-y^{\,\prime }-12y=0\).
(3)
A 1st order differential equation \(y'=f(t,y)\) with a so-called \(``\)initial condition" \(y(t_0)=y_0\) is called an Initial Value Problem (abbreviated \(``\)IVP"). An \(``\)IVP solution" must satisfy both the differential equation and the initial condition. There is a unique solution \(y(t)\) to the 1st order IVP \(ty^{\,\prime }-y=-\dfrac {200}{t}\), \(y(1)=101\) on the interval \((0,\infty )\). Find \(y(2026)\) to the nearest hundredth.
(4)
On which of the following open intervals is it guaranteed that there is NO solution to the IVP \((x+1)y^{\,\prime }+y=1\), \(y(0)=2\)? Select all that apply.
(a)
\((-\infty ,\infty )\)
(b)
\((-1,1)\)
(c)
\((-1,\infty )\)
(d)
\((-\infty ,1)\)
(e)
\((-2,1)\)
(f)
\((-2,2)\)
(g)
\((-2,-1)\)
(h)
\((-1,2)\)
(5)
Suppose an epidemic spreading through a certain population is modeled by the differential equation \(y^{\,\prime }(t)=ay(t)-b[y(t)]^2\) where \(a,b\) are positive constants and \(y(t)\) is the number of individuals who have the disease at time \(t\) days. Below is a direction field for such a model. Suppose we are given that \(a=0.75\) and \(\displaystyle \lim _{t\rightarrow \infty } y(t)=6,000\) for any non-trivial solution \(y(t)\). Determine \(b\). Round to the nearest ten-thousandth.
A direction field
Figure 2: Another direction field