Exercises

(1)
Which of the following equations are Bernoulli equations? Select all that apply.
(a)
\(y^{\,\prime }=xy\)
(b)
\(y^{\,\prime }=xy+y^2\)
(c)
\(y^{\,\prime }=x^2y+x\)
(d)
\(y^{\,\prime }=x^2y+y\)
(e)
\(y^{\,\prime }=y+x\sqrt {y}\)
(f)
\(y^{\,\prime }=x^2y+y^2+x\)
(g)
\(y^{\,\prime }=x^2y+y+\dfrac {1}{y}\)
(h)
\(y^{\,\prime }=e^x y +x e^y\)
(i)
\(y^{\,\prime }=\dfrac {xy^2+1}{y}\)
(2)
The equation \(2xy^4\,y^{\,\prime }+\dfrac {x^3}{y^2}=x^2y^5\) is a Bernoulli equation, but is not in the standard form for a Bernoulli equation, which is \(y^{\,\prime }+p(x)y=f(x)y^r\) for some real number \(r\neq 0,1\). Write this Bernoulli equation in the standard form and determine the value of \(r\).
(3)
Suppose \(y(x)\) satisfies the Bernoulli IVP \(xy^{\,\prime }=y-3xy^3\), \(y(1)=1\). Find \(y(2026)\) to the nearest ten-thousandth.
(4)
Suppose \(y(x)\) satisfies the Bernoulli IVP \(xy^{\,\prime }=y-3xy^3\), \(y(1)=0\). Find \(y(2026)\) to the nearest ten-thousandth.
(5)
Some models for population growth use differential equations of the form \(p^{\,\prime }(t)=aP(t)\left [1-g(t)P(t)\right ]\) where \(P(t)\) is the population size at time \(t\geq 0\), \(a\) is a positive constant, and \(g(t)\) is a continuous function of \(t\) on \([0,\infty )\). Assuming \(P(0)>0\) and \(\displaystyle \lim _{t\rightarrow \infty } e^{-at} \int _0^t g(x)e^{ax}\;dx=L>0\) find \(\displaystyle \lim _{t\rightarrow \infty } P(t)\) in terms of \(a,L\).
(a)
\(0\)
(b)
\(La\)
(c)
\(\dfrac {L}{a}\)
(d)
\(\dfrac {a}{L}\)
(e)
\(\dfrac {1}{La}\)
(f)
\(La^2\)
(g)
\(L^2a\)
(h)
\(L^2a^2\)