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Chapter 8: Problem 17

Find the general solution of each differential equation. Use \(C, C_{1}, C_{2}, \ldots .\) to denote arbitrary constants. $$y^{\prime}(x)=4 \tan 2 x-3 \cos x$$

Short Answer

Expert verified
Answer: The general solution for the given differential equation is \(y(x) = 2\ln|\sec(2x)| - 3\sin(x) + C\), where C is an arbitrary constant.

Step by step solution

01

Identify the equation type

The given differential equation is in the form: $$y'(x) = 4\tan(2x) - 3\cos(x)$$ This is a first-order, non-homogeneous differential equation with the derivative of the function only.
02

Integrate the equation

To find the general solution y(x), we have to integrate the given equation: $$y(x) = \int (4\tan(2x) - 3\cos(x)) dx$$
03

Integrate each term separately

We can integrate each term separately: $$y(x) = 4\int \tan(2x) dx - 3\int \cos(x) dx$$
04

Integrate the tangent function

Use the substitution method for integrating the tangent function, $$u = 2x$$ $$du = 2 dx$$ and replace variable x in the integral. Then divide by the constant from the substitution: $$4\int \tan(2x) dx = 2\int \tan(u) du$$ Now integrate \(\tan(u)\) function: $$2\int \tan(u) du = 2\ln| \sec(u) | + C_1$$ Replace u with the original variable x: $$2\ln| \sec(2x) | + C_1$$
05

Integrate the cosine function

Now let's integrate the second term: $$3 \int \cos(x) dx = 3\sin(x) + C_2$$
06

Combine the results

Combine the results from the integration of the first and second terms and write the general solution as follows: $$y(x) = 2\ln|\sec(2x)| - 3\sin(x) + C$$ Here, C is the arbitrary constant resulting from the combination of \(C_1\) and \(C_2\). The general solution for the given first-order non-homogeneous differential equation is: $$y(x) = 2\ln|\sec(2x)| - 3\sin(x) + C$$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

First-Order Differential Equations
A first-order differential equation is one of the simplest forms of differential equations. These equations involve the first derivative of a function. In the simplest terms, it tells us how a function is changing at a given point. The general form of a first-order differential equation is expressed as \[ y'(x) = f(x, y). \]Here, \( y' \) denotes the derivative of \( y \) with respect to \( x \), and \( f(x, y) \) is a given function.
  • First-order equations can be linear or non-linear.
  • They are crucial in modeling real-world processes that change over time.
In our exercise, the equation \( y'(x) = 4\tan(2x) - 3\cos(x) \) represents a first-order form. Understanding such equations is fundamental in calculus, particularly when dealing with changes in natural and engineered systems.
Integration Techniques
Integration is a key process in solving differential equations. It helps us find the original function when we know its rate of change. There are various integration techniques that can be employed, such as substitution, integration by parts, and partial fractions.In the step-by-step solution, two main techniques are used:
  • Substitution Technique: Used for the integration of \( \tan(2x). \) By substituting \( u = 2x, \) we simplify the integration process.
  • Direct Integration: For \( \cos(x), \) the integral is straightforward, resulting in \( \sin(x). \)
Integrating each term of the differential equation separately allows us to solve complex equations piece by piece.
Non-Homogeneous Differential Equations
A non-homogeneous differential equation has terms that make it more complex than homogeneous equations. Homogeneous equations equal zero, while non-homogeneous ones equal a function other than zero or involve terms without derivatives.The equation in this problem, \( y'(x) = 4\tan(2x) - 3\cos(x), \) is non-homogeneous because of the terms \( 4\tan(2x) \) and \( -3\cos(x). \)
  • Non-homogenous equations: Often require special solution techniques because they include additional functions on the right-hand side.
  • Finding a particular solution helps to account for these extra terms.
Recognizing and solving non-homogeneous equations allows us to model more complex situations where factors change over time, providing deeper insights into dynamics where simple models are insufficient.

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Most popular questions from this chapter

Determine whether the following equations are separable. If so, solve the initial value problem. $$\frac{d y}{d x}=e^{x-y}, y(0)=\ln 3$$

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. If the growth rate function for a population model is positive, then the population is increasing. b. The solution of a stirred tank initial value problem always approaches a constant as \(t \rightarrow \infty\) c. In the predator-prey models discussed in this section, if the initial predator population is zero and the initial prey population is positive, then the prey population increases without bound.

Solve the following initial value problems and leave the solution in implicit form. Use graphing software to plot the solution. If the implicit solution describes more than one curve, be sure to indicate which curve corresponds to the solution of the initial value problem. $$y^{\prime}(x)=\sqrt{\frac{x+1}{y+4}}, y(3)=5$$

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.

Consider the following pairs of differential equations that model a predator- prey system with populations \(x\) and \(y .\) In each case, carry out the following steps. a. Identify which equation corresponds to the predator and which corresponds to the prey. b. Find the lines along which \(x^{\prime}(t)=0 .\) Find the lines along which \(y^{\prime}(t)=0\) c. Find the equilibrium points for the system. d. Identify the four regions in the first quadrant of the xy-plane in which \(x^{\prime}\) and \(y^{\prime}\) are positive or negative. e. Sketch a representative solution curve in the xy-plane and indicate the direction in which the solution evolves. $$x^{\prime}(t)=2 x-x y, y^{\prime}(t)=-y+x y$$
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