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Chapter 8: Problem 3

What is the general solution of the equation \(y^{\prime}(t)=-4 y+6 ?\)

Short Answer

Expert verified
Answer: The general solution of the given ODE is \(y(t)= \frac{3}{2} + Ce^{-4t}\), where \(C\) is the constant of integration.

Step by step solution

01

Identify the integrating factor

The given ODE can be expressed as \(\frac{dy}{dt} + 4y = 6\). In this case, the function \(P(t)\) is constant and equal to \(4\). Therefore, the integrating factor is given by \(I(t) = e^{\int P(t) dt}\). Let's calculate the integrating factor: \(I(t) = e^{\int 4 dt} = e^{4t}\)
02

Multiply the whole equation by the integrating factor

Now, we will multiply the entire equation by the integrating factor \(I(t) = e^{4t}\): \(e^{4t} \frac{dy}{dt} + 4e^{4t}y = 6e^{4t}\)
03

Integrate both sides of the equation

Observe that the left side of the resulting equation is the derivative of the product of \(y(t)\) and the integrating factor \(I(t)\), i.e., \(\frac{d}{dt} (ye^{4t})\). Now, we will integrate both sides of the equation with respect to \(t\): \(\int \frac{d}{dt}(ye^{4t}) dt = \int 6e^{4t} dt\) Integrating both sides, we obtain: \(ye^{4t} = \frac{3}{2}e^{4t} + C\), where \(C\) is the constant of integration.
04

Write the general solution

Now, we will isolate \(y(t)\): \(y(t) = \frac{3}{2} + Ce^{-4t}\) Hence, the general solution of the given equation is \(y(t) = \frac{3}{2} + Ce^{-4t}\).

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Most popular questions from this chapter

Use the method outlined in Exercise 43 to solve the following Bernoulli equations. a. \(y^{\prime}(t)+y=2 y^{2}\) b. \(y^{\prime}(t)-2 y=3 y^{-1}\) c. \(y^{\prime}(t)+y=\sqrt{y}\)

a. Show that for general positive values of \(R, V, C_{i},\) and \(m_{0},\) the solution of the initial value problem $$m^{\prime}(t)=-\frac{R}{V} m(t)+C_{i} R, \quad m(0)=m_{0}$$ is \(m(t)=\left(m_{0}-C_{i} V\right) e^{-R t / V}+C_{i} V\) b. Verify that \(m(0)=m_{0}\) c. Evaluate \(\lim m(t)\) and give a physical interpretation of the result. d. Suppose \(^{t} \vec{m}_{0}^{\infty}\) and \(V\) are fixed. Describe the effect of increasing \(R\) on the graph of the solution.

Determine whether the following equations are separable. If so, solve the initial value problem. $$y^{\prime}(t)=y\left(4 t^{3}+1\right), y(0)=4$$

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 \(3 \mathrm{L} / \mathrm{min}\)

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.
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