Question

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

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

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.

Integrate the equation

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

Integrate each term separately

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

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: $$2\ln| \sec(2x) | + C_1$$

Integrate the cosine function

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

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: $$y(x) = 2\ln|\sec(2x)| - 3\sin(x) + C$$

Key concepts

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.