Problem 27
Solve the initial value problem $$ y^{\prime}=a y-b y^{2}, \quad y(0)=y_{0} $$ Discuss the behavior of the solution if (a) \(y_{0} \geq 0 ;\) (b) \(y_{0}<0\).
Problem 27
Solve the initial value problem. $$ x y y^{\prime}=3 x^{2}+4 y^{2}, \quad y(1)=\sqrt{3} $$
Problem 28
In Exercises \(25-29\) solve the initial value problem and sketch the graph of the solution. $$ \text { C/G } y^{\prime}+(\cot x) y=\cos x, \quad y\left(\frac{\pi}{2}\right)=1 $$
Problem 28
Suppose \(a, b, c,\) and \(d\) are constants such that \(a d-b c \neq 0,\) and let \(m\) and \(n\) be arbitrary real numbers. Show that $$ \left(a x^{m} y+b y^{n+1}\right) d x+\left(c x^{m+1}+d x y^{n}\right) d y=0 $$ has an integrating factor \(\mu(x, y)=x^{\alpha} y^{\beta}\).
Problem 28
(a) Solve the exact equation $$ \left(x^{2}+y^{2}\right) d x+2 x y d y=0 $$ implicitly, (b) For what choices of \(\left(x_{0}, y_{0}\right)\) does Theorem 2.3 .1 imply that the initial value problem $$ \left(x^{2}+y^{2}\right) d x+2 x y d y=0, \quad y\left(x_{0}\right)=y_{0}, $$ has a unique solution \(y=y(x)\) on some open interval \((a, b)\) that contains \(x_{0} ?\) (c) Plot a direction field and some integral curves for (A). From the plot determine, the interval \((a, b)\) of (b), the monotonicity properties (if any) of the solution of (B), and \(\lim _{x \rightarrow a+} y(x)\) and \(\lim _{x \rightarrow b-} y(x) .\) HiNT: Your answers will depend upon which quadrant contains \(\left(x_{0}, y_{0}\right)\).
Problem 28
In Exercises \(28-34\) solve the given homogeneous equation implicitly. $$ y^{\prime}=\frac{x+y}{x-y} $$
Problem 28
The population \(P=P(t)\) of a species satisfies the logistic equation $$ P^{\prime}=a P(1-\alpha P) $$ and \(P(0)=P_{0}>0 .\) Find \(P\) for \(t>0,\) and find \(\lim _{t \rightarrow \infty} P(t)\)
Problem 29
Solve the given homogeneous equation implicitly. $$ \left(y^{\prime} x-y\right)(\ln |y|-\ln |x|)=x $$
Problem 29
Suppose \(M, N, M_{x},\) and \(N_{y}\) are continuous for all \((x, y),\) and \(\mu=\mu(x, y)\) is an integrating factor for $$ M(x, y) d x+N(x, y) d y=0. $$ Assume that \(\mu_{x}\) and \(\mu_{y}\) are continuous for all \((x, y),\) and suppose \(y=y(x)\) is a differentiable function such that \(\mu(x, y(x))=0\) and \(\mu_{x}(x, y(x)) \neq 0\) for all \(x\) in some interval \(I\). Show that \(y\) is a solution of \((\mathrm{A})\) on \(I\)
Problem 29
An epidemic spreads through a population at a rate proportional to the product of the number of people already infected and the number of people susceptible, but not yet infected. Therefore, if \(S\) denotes the total population of susceptible people and \(I=I(t)\) denotes the number of infected people at time \(t,\) then $$ I^{\prime}=r I(S-I) $$ where \(r\) is a positive constant. Assuming that \(I(0)=I_{0},\) find \(I(t)\) for \(t>0,\) and show that \(\lim _{t \rightarrow \infty} I(t)=S\)
