Question

Finding general solutions 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

Question: Find the general solution of the first-order differential equation given by $$y'(x) = 4\tan(2x) - 3\cos(x).$$ Answer: The general solution of 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

Integrate the given equation with respect to x

We will integrate the given equation with respect to x: $$y(x) = \int(4\tan(2x) - 3\cos(x))\,dx.$$

Break the integration into two parts

Now, we can break the integration into two separate parts: $$y(x) = \int4\tan(2x)\,dx - \int3\cos(x)\,dx.$$

Integrate the first part: \(\int4\tan(2x)\,dx\)

We can integrate \(4\tan(2x)\) with respect to \(x\) by using the substitution method. Let: $$u = 2x \Rightarrow \frac{du}{dx} = 2 \Rightarrow dx = \frac{1}{2}du.$$ Now we substitute into the integration: $$\int4\tan(2x)\,dx= 4\int\tan(u)\, \frac{1}{2}\,du= 2\int\tan(u)\,du.$$ Now, integrating \(\tan(u)\), we have: $$2\int\tan(u)\,du = 2(\ln|\sec(u)|+C_1) = 2\ln|\sec(2x)|+C_1$$

Integrate the second part: \(\int3\cos(x)\,dx\)

The integration of the second part is straightforward: $$\int 3\cos(x)\,dx = 3 \sin(x) + C_2.$$

Combine the results of Step 3 and Step 4

Now, we will combine the results of Step 3 and Step 4: $$y(x) = 2\ln|\sec(2x)|+C_1 + 3 \sin(x) + C_2.$$

Write the general solution

Finally, we will write the general solution of the given differential equation by combining the arbitrary constants \(C_1\) and \(C_2\) into a single constant \(C\): $$y(x) = 2\ln|\sec(2x)| + 3\sin(x) + C.$$

Key concepts

Integration

Integration is a fundamental concept in calculus, crucial for solving differential equations. It involves finding a function, called the antiderivative, whose derivative is the integrand (the function being integrated).
When we find the integral of a function, we're essentially performing the reverse of differentiation.
This process allows us to accumulate quantities, such as areas under curves or solutions to differential equations. There are several integration methods, and selecting the most suitable one is often crucial:

• Indefinite Integrals: These integrals don't have specified limits and include an arbitrary constant of integration, often denoted by 'C'.
• Definite Integrals: These include limits of integration, providing a numerical value representing the accumulated change between the limits.

Breaking larger integrals into manageable parts, as seen in the original exercise, is a common technique to simplify the process.

General Solution

In the context of differential equations, a general solution encompasses all possible solutions through the inclusion of arbitrary constants.
These solutions reveal the behavior of a system or phenomenon described by the differential equation. General solutions are composed of two components:

• Particular Solution: A specific solution that satisfies the differential equation without arbitrary constants.
• Arbitrary Constants: Symbols like 'C', 'C_1', and 'C_2' that account for the initial conditions or specific circumstances of the problem.

The arbitrary constants in the general solution provide flexibility, allowing the user to tailor it according to initial conditions or further constraints.
This adaptability is key when applying theoretical solutions to practical scenarios.

Substitution Method

The substitution method is a powerful tool used in calculus to simplify integration, making it more manageable.
This technique involves introducing a new variable to substitute a part of the original integrand, thus transforming the integral into a simpler form.Here's how it typically works:

• Choose a Substitution: Identify a part of the integrand that complicates direct integration.
• Define a New Variable: Let this part of the integrand be equal to a new variable, such as 'u'.
  Express 'dx' in terms of 'du' to facilitate substitution.
• Transform and Integrate: Rewrite the integral in terms of the new variable, integrate, then revert to the original variable.

In our example, substituting 'u = 2x' simplifies the integral of '4\( \tan(2x) \)', enabling us to integrate more easily, focusing on constant factors like '2'.

Arbitrary Constants

Arbitrary constants are a crucial part of the solution in differential equations and integrals.
These constants, denoted typically by 'C', provide the means to incorporate different particular solutions under one general framework. The need for arbitrary constants arises from the nature of integration:

• Integration's Nature: Each indefinite integral contains a constant of integration because differentiating a constant yields zero.
  This means any constant added during integration yields the same derivative.
• Defining Initial Conditions: Arbitrary constants are pivotal in tailoring the solution to satisfy initial conditions.
  This allows us to determine a particular solution from the general one.

In the context of differential equations, merging multiple constants (like 'C_1' and 'C_2') into a single 'C' still represents all valid solutions, preserving generality while simplifying the expression.