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Chapter 4: Problem 74

Find the solution of the following initial value problems. $$u^{\prime}(x)=\frac{e^{2 x}+4 e^{-x}}{e^{x}} ; u(\ln 2)=2$$

Short Answer

Expert verified
Question: Find the particular solution to the initial value problem (IVP) \(u'(x) = \frac{e^{2x} + 4e^{-x}}{e^x}\) with \(u(\ln 2) = 2\). Answer: The particular solution to the given IVP is \(u(x) = e^x - 2e^{-2x} + \frac{1}{2}\).

Step by step solution

01

Write the equation

Rewrite the given differential equation: $$u'(x) = \frac{e^{2x} + 4e^{-x}}{e^x}$$
02

Integrate both sides

To obtain the general solution, integrate both sides of the equation with respect to x: $$\int u'(x) dx = \int \frac{e^{2x} + 4e^{-x}}{e^x}dx$$
03

Simplify the integrand

Simplify the integrand by dividing the terms inside the integral by \(e^x\): $$\int u'(x) dx = \int \left(e^x + 4e^{-2x}\right)dx$$
04

Integrate the simplified equation

Now, we can easily integrate the simplified equation: $$u(x) = \int \left(e^x + 4e^{-2x}\right)dx = e^x - 2e^{-2x} + C$$ Here, \(C\) is the constant of integration.
05

Apply the initial condition

To find the particular solution, apply the initial condition \(u(\ln 2) = 2\): $$2 = e^{\ln 2} - 2e^{-2\ln 2} + C$$
06

Calculate the constant of integration

Calculate the value of \(C\): $$2 = 2 - \frac{1}{2} + C$$ $$C = \frac{1}{2}$$
07

Write the particular solution

Finally, substitute the value of \(C\) back into the general solution to obtain the particular solution: $$u(x) = e^x - 2e^{-2x} + \frac{1}{2}$$

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Most popular questions from this chapter

The population of a species is given by the function \(P(t)=\frac{K t^{2}}{t^{2}+b},\) where \(t \geq 0\) is measured in years and \(K\) and \(b\) are positive real numbers. a. With \(K=300\) and \(b=30,\) what is \(\lim P(t),\) the carrying capacity of the population? b. With \(K=300\) and \(b=30,\) when does the maximum growth rate occur? c. For arbitrary positive values of \(K\) and \(b,\) when does the maximum growth rate occur (in terms of \(K\) and \(b\) )?

The graph of \(f^{\prime}\) on the interval [-3,2] is shown in the figure. a. On what interval(s) is \(f\) increasing? Decreasing? b. Find the critical points of \(f .\) Which critical points correspond to local maxima? Local minima? Neither? c. At what point(s) does \(f\) have an inflection point? d. On what interval(s) is \(f\) concave up? Concave down? e. Sketch the graph of \(f^{\prime \prime}\) f. Sketch one possible graph of \(f\)

Verify the following indefinite integrals by differentiation. $$\int \frac{x}{\left(x^{2}-1\right)^{2}} d x=-\frac{1}{2\left(x^{2}-1\right)}+C$$

Locate the critical points of the following functions and use the Second Derivative Test to determine whether they correspond to local maxima, local minima, or neither. $$p(t)=2 t^{3}+3 t^{2}-36 t$$

Sketch the graph of a function that is continuous on \((-\infty, \infty)\) and satisfies the following sets of conditions. $$\begin{array}{l}f^{\prime \prime}(x)>0 \text { on }(-\infty,-2) ; f^{\prime \prime}(x)<0 \text { on }(-2,1) ; f^{\prime \prime}(x)>0 \text { on } \\\\(1,3) ; f^{\prime \prime}(x)<0 \text { on }(3, \infty)\end{array}$$
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