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Chapter 8: Problem 13

Solve the following initial value problems. $$y^{\prime}(t)-2 y=8, y(0)=0$$

Short Answer

Expert verified
Question: Determine the particular solution for the given initial value problem involving a first-order linear differential equation: $y'(t) - 2y(t) = 8$, with an initial condition $y(0) = 0$. Answer: The particular solution for the given initial value problem is $y(t) = -4 + 4e^{2t}$.

Step by step solution

01

Identify the type of differential equation

This is a first-order linear differential equation in the form: $$y'(t) + P(t)y(t) = Q(t)$$ Where \(P(t) = -2\) and \(Q(t) = 8\).
02

Find the integrating factor

We need to find the integrating factor, which is given by: $$\mu(t) = e^{\int P(t) dt}$$ In our case, \(P(t) = -2\), so we have: $$\mu(t) = e^{\int -2 dt} = e^{-2t}$$
03

Multiply both sides by the integrating factor

Multiply both sides of the equation by the integrating factor \(\mu(t) = e^{-2t}\): $$(e^{-2t}y'(t)) - 2e^{-2t}y(t) = 8e^{-2t}$$ The left side of this equation is now the derivative of the product \(\mu(t)y(t)\): $$\frac{d}{dt}(e^{-2t}y(t)) = 8e^{-2t}$$
04

Integrate both sides with respect to t

Now, we integrate both sides with respect to \(t\): $$\int\frac{d}{dt}(e^{-2t}y(t)) dt = \int 8e^{-2t} dt$$ This yields: $$e^{-2t}y(t) = -4e^{-2t} + C$$
05

Solve for y(t)

Now, we solve for \(y(t)\) by multiplying both sides by \(e^{2t}\): $$y(t) = -4 + Ce^{2t}$$
06

Apply the initial condition to find the particular solution

Finally, to find the particular solution, substitute the initial condition \(y(0) = 0\) into the form of the general solution: $$0 = -4 + Ce^{2 \times 0} \Rightarrow 0 = -4 + C$$ This gives \(C = 4\), so the particular solution is: $$y(t) = -4 + 4e^{2t}$$

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Most popular questions from this chapter

One possible model that describes the free fall of an object in a gravitational field subject to air resistance uses the equation \(v^{\prime}(t)=g-b v,\) where \(v(t)\) is the velocity of the object for \(t \geq 0\), \(g=9.8 \mathrm{m} / \mathrm{s}^{2}\) is the acceleration due to gravity, and \(b>0\) is a constant that involves the mass of the object and the air resistance. a. Verify by substitution that a solution of the equation, subject to the initial condition \(v(0)=0,\) is \(v(t)=\frac{g}{b}\left(1-e^{-b t}\right)\). b. Graph the solution with \(b=0.1 s^{-1}\). c. Using the graph in part (c), estimate the terminal velocity \(\lim _{t \rightarrow \infty} v(t)\).

Consider the initial value problem \(y^{\prime}(t)=y^{n+1}, y(0)=y_{0},\) where \(n\) is a positive integer. a. Solve the initial value problem with \(n=1\) and \(y_{0}=1\) b. Solve the initial value problem with \(n=2\) and \(y_{0}=\frac{1}{\sqrt{2}}\) c. Solve the problem for positive integers \(n\) and \(y_{0}=n^{-1 / n}\) How do solutions behave as \(t \rightarrow 1^{-2}\)

Let \(y(t)\) be the population of a species that is being harvested, for \(t \geq 0 .\) Consider the harvesting model \(y^{\prime}(t)=0.008 y-h, y(0)=y_{0},\) where \(h\) is the annual harvesting rate, \(y_{0}\) is the initial population of the species, and \(t\) is measured in years. a. If \(y_{0}=2000,\) what harvesting rate should be used to maintain a constant population of \(y=2000,\) for \(t \geq 0 ?\) b. If the harvesting rate is \(h=200 /\) year, what initial population ensures a constant population?

A differential equation of the form \(y^{\prime}(t)=f(y)\) is said to be autonomous (the function \(f\) depends only on \(y\) ). The constant function \(y=y_{0}\) is an equilibrium solution of the equation provided \(f\left(y_{0}\right)=0\) (because then \(y^{\prime}(t)=0\) and the solution remains constant for all \(t\) ). Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations. a. Find the equilibrium solutions. b. Sketch the direction field, for \(t \geq 0\). c. Sketch the solution curve that corresponds to the initial condition \(y(0)=1\). $$y^{\prime}(t)=y(y-3)$$

Determine whether the following equations are separable. If so, solve the initial value problem. $$y^{\prime}(t)=e^{t y}, y(0)=1$$
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