/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 11 Solve the following initial valu... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 11

Solve the following initial value problems. $$y^{\prime}(t)=3 y-6, y(0)=9$$

Short Answer

Expert verified
Question: Determine the function y(t) that satisfies the initial value problem for the given first-order differential equation and initial condition: $\frac{dy}{dt} = 3y - 6$, $y(0) = 9$. Answer: The function y(t) that satisfies the initial value problem is $y(t) = y - 2 + 11e^{-3t}$.

Step by step solution

01

Rewrite the Given Equation

Rewrite the given equation to emphasize that it is a first-order differential equation: $$\frac{dy}{dt} = 3y - 6$$
02

Solve the Differential Equation

Apply the Integrating Factor method to solve the differential equation. Define the integrating factor, IF, as: $$ IF = e^{\int P(t) dt }$$ Where P(t) is the coefficient of y in the given equation. In our case, P(t) = 3. Calculate the integral of P(t): $$\int P(t) dt = \int 3 dt = 3t$$ Find the integrating factor: $$IF = e^{3t}$$ Multiply both sides of the given differential equation by the integrating factor: $$e^{3t}\frac{dy}{dt} = 3e^{3t}y - 6e^{3t}$$ The left side of the equation is now an exact differential: $$\frac{d}{dt}(e^{3t}y) = 3e^{3t}y - 6e^{3t}$$ Integrate both sides with respect to t: $$\int \frac{d}{dt}(e^{3t}y)dt = \int (3e^{3t}y - 6e^{3t})dt$$ $$e^{3t}y = e^{3t}y - 2e^{3t} + C$$
03

Solve for y(t)

To find the general solution, solve for y(t): $$y(t) = e^{-3t}(e^{3t}y - 2e^{3t} + C)$$ $$y(t) = y - 2 + Ce^{-3t}$$
04

Apply the Initial Condition

Now, use the initial condition y(0) = 9 to find the value of the constant C: $$9 = y(0) = y - 2 + Ce^{-3*0}$$ $$9 = y - 2 + C$$ $$C = 11$$
05

Write the Particular Solution

Substitute the value of C back into the general solution to find the particular solution for the given initial value problem: $$y(t) = y - 2 + 11e^{-3t}$$ The solution to the initial value problem is: $$y(t) = y - 2 + 11e^{-3t}$$

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Most popular questions from this chapter

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 \(y(0)=1\). $$y^{\prime}(t)=\sin y$$

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

Determine whether the following statements are true and give an explanation or counterexample. a. The general solution of \(y^{\prime}(t)=2 y-18\) is \(y(t)=2 e^{2 t}+9\) b. If \(k>0\) and \(b>0,\) then \(y(t)=0\) is never a solution of \(y^{\prime}(t)=k y-b\) c. The equation \(y^{\prime}(t)=t y(t)+3\) is separable and can be solved using the methods of this section. d. According to Newton's Law of Cooling, the temperature of a hot object will reach the ambient temperature after a finite amount of time.

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 \(t \geq 0\).

The reaction of certain chemical compounds can be modeled using a differential equation of the form \(y^{\prime}(t)=-k y^{n}(t),\) where \(y(t)\) is the concentration of the compound for \(t \geq 0, k>0\) is a constant that determines the speed of the reaction, and \(n\) is a positive integer called the order of the reaction. Assume that the initial concentration of the compound is \(y(0)=y_{0}>0\). a. Consider a first-order reaction \((n=1)\) and show that the solution of the initial value problem is \(y(t)=y_{0} e^{-k t}\). b. Consider a second-order reaction \((n=2)\) and show that the solution of the initial value problem is \(y(t)=\frac{y_{0}}{y_{0} k t+1}\). c. Let \(y_{0}=1\) and \(k=0.1 .\) Graph the first-order and secondorder solutions found in parts (a) and (b). Compare the two reactions.
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