Chapter 8: Problem 2
Is the equation \(t^{2} y^{\prime}(t)=\frac{t+4}{y^{2}}\) separable?
Over 30 million students worldwide already upgrade their learning with Vaia!
All the tools & learning materials you need for study success - in one app.
Get started for free
Solve the following initial value problems and leave the solution in implicit form. Use graphing software to plot the solution. If the implicit solution describes more than one curve, be sure to indicate which curve corresponds to the solution of the initial value problem. $$z^{\prime}(x)=\frac{z^{2}+4}{x^{2}+16}, z(4)=2$$
Consider the first-order initial value problem \(y^{\prime}(t)=a y+b, y(0)=A,\) for \(t \geq 0,\) where \(a, b,\) and \(A\) are real numbers. a. Explain why \(y=-b / a\) is an equilibrium solution and corresponds to a horizontal line in the direction field. b. Draw a representative direction field in the case that \(a>0\) Show that if \(A>-b / a,\) then the solution increases for \(t \geq 0\) and if \(A<-b / a,\) then the solution decreases for \(t \geq 0\). c. Draw a representative direction field in the case that \(a<0\) Show that if \(A>-b / a,\) then the solution decreases for \(t \geq 0\) and if \(A<-b / a,\) then the solution increases for \(t \geq 0\).
Verify that the function $$M(t)=K\left(\frac{M_{0}}{K}\right)^{\exp (-r t)}$$ satisfies the properties \(M(0)=M_{0}\) and \(\lim _{t \rightarrow \infty} M(t)=K\).
Determine whether the following equations are separable. If so, solve the initial value problem. $$\frac{d y}{d x}=e^{x-y}, y(0)=\ln 3$$
Consider the differential equation \(y^{\prime}(t)=\frac{y(y+1)}{t(t+2)}\) and carry out the following analysis. a. Show that the general solution of the equation can be written in the form $$ y(t)=\frac{\sqrt{t}}{C \sqrt{t+2}-\sqrt{t}} $$ b. Now consider the initial value problem \(y(1)=A,\) where \(A\) is a real number. Show that the solution of the initial value problem is $$ y(t)=\frac{\sqrt{t}}{\left(\frac{1+A}{\sqrt{3} A}\right) \sqrt{t+2}-\sqrt{t}} $$ c. Find and graph the solution that satisfies the initial condition \(y(1)=1\) d. Describe the behavior of the solution in part (c) as \(t\) increases. e. Find and graph the solution that satisfies the initial condition \(y(1)=2\) f. Describe the behavior of the solution in part (e) as \(t\) increases. g. In the cases in which the solution is bounded for \(t>0,\) what is the value of \(\lim _{t \rightarrow \infty} y(t) ?\)
What do you think about this solution?
We value your feedback to improve our textbook solutions.
