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Chapter 8: Problem 1

What is a separable first-order differential equation?

Short Answer

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Question: Determine if the following first-order differential equation is separable: \[\frac{dy}{dt} = y^2 + 3t\] Answer: In order to determine if this differential equation is separable, we need to rewrite it in the form \[\frac{dy}{dt} = g(t)h(y)\]. However, we cannot find continuous functions \(g(t)\) and \(h(y)\) that would result in the equation \[\frac{dy}{dt} = g(t)h(y)\] having the same form as our original equation, \[\frac{dy}{dt} = y^2 + 3t\]. Therefore, this differential equation is not separable.

Step by step solution

01

Definition of a Separable First-order Differential Equation

A first-order differential equation is said to be separable if it can be written in the form of \[\frac{dy}{dt} = g(t)h(y)\] where \(g(t)\) and \(h(y)\) are continuous functions of \(t\) and \(y\), respectively. In other words, a separable differential equation can be separated into parts consisting of functions dependent on \(y\) alone and functions dependent on \(t\) alone.
02

Recognizing a Separable Differential Equation

To determine if a given first-order differential equation is separable, try to rewrite it in the form \[\frac{dy}{dt} = g(t)h(y)\]. If this can be done, then the differential equation is separable. Note that sometimes, you may need to manipulate the equation algebraically to achieve this form.
03

Example of a Separable Differential Equation

Consider the following first-order differential equation: \[\frac{dy}{dt} = ty (1-y)\] This differential equation has the structure of a separable equation, as we can rewrite it as: \[\frac{dy}{dt} = t(1-y)y\] Here, the functions \(g(t) = t\) and \(h(y) = (1-y)y\). Thus, this is a separable first-order differential equation.

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Most popular questions from this chapter

For each of the following stirred tank reactions, carry out the following analysis. a. Write an initial value problem for the mass of the substance. b. Solve the initial value problem and graph the solution to be sure that \(m(0)\) and \(\lim _{t \rightarrow \infty} m(t)\) are correct. A \(1500-L\) tank is initially filled with a solution that contains \(3000 \mathrm{g}\) of salt. A salt solution with a concentration of \(20 \mathrm{g} / \mathrm{L}\) flows into the tank at a rate of \(3 \mathrm{L} / \mathrm{min.}\) The thoroughly mixed solution is drained from the tank at a rate of \(3 \mathrm{L} / \mathrm{min}\)

A special class of first-order linear equations have the form \(a(t) y^{\prime}(t)+a^{\prime}(t) y(t)=f(t),\) where \(a\) and \(f\) are given functions of \(t.\) Notice that the left side of this equation can be written as the derivative of a product, so the equation has the form $$a(t) y^{\prime}(t)+a^{\prime}(t) y(t)=\frac{d}{d t}(a(t) y(t))=f(t)$$ Therefore, the equation can be solved by integrating both sides with respect to \(t.\) Use this idea to solve the following initial value problems. $$e^{-t} y^{\prime}(t)-e^{-t} y=e^{2 t}, y(0)=4$$

Write a logistic equation with the following parameter values. Then solve the initial value problem and graph the solution. Let \(r\) be the natural growth rate, \(K\) the carrying capacity, and \(P_{0}\) the initial population. $$r=0.2, K=300, P_{0}=50$$

Solution of the logistic equation Use separation of variables to show that the solution of the initial value problem $$P^{\prime}(t)=r P\left(1-\frac{P}{K}\right), \quad P(0)=P_{0}$$ $$\text { is } P(t)=\frac{K}{\left(\frac{K}{P_{0}}-1\right) e^{-n}+1}$$

Determine whether the following equations are separable. If so, solve the initial value problem. $$y^{\prime}(t)=\frac{e^{t}}{2 y}, y(\ln 2)=1$$
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