Problem 57 - Calculus for Biology and Medicine

Question

In Problems 47-58, find the general solution of the differential equation. $$ \frac{d y}{d x}=\sec ^{2}\left(\frac{x}{2}\right),-1

Step-by-Step Solution

Answer

The general solution is \( y = 2 \tan \left( \frac{x}{2} \right) + C \).

Step 1: Identify the Differential Equation

The given differential equation is \( \frac{d y}{d x} = \sec^2\left(\frac{x}{2}\right) \). This is a first-order differential equation already separated because \( y \) is not on the right side.

Step 2: Integrate Both Sides

To find the solution, integrate both sides with respect to \( x \). The equation becomes \[ \int d y = \int \sec^2\left(\frac{x}{2}\right) d x \].

Step 3: Solve the Integral on the Right

The integral \( \int \sec^2\left(\frac{x}{2}\right) d x \) requires a substitution. Let \( u = \frac{x}{2} \); hence, \( du = \frac{1}{2} dx \) or \( dx = 2du \). The integral becomes \[ \int \sec^2(u) 2 du = 2 \int \sec^2(u) du \], which evaluates to \( 2\tan(u) + C \). Return the expression in terms of \( x \) to get \( 2\tan\left(\frac{x}{2}\right) + C \).

Step 4: Write the General Solution

The integration gives \( y = 2\tan\left(\frac{x}{2}\right) + C \), where \( C \) is the constant of integration. This represents the general solution for the differential equation.

Key Concepts

General Solution

A **general solution** of a differential equation encompasses all possible solutions that satisfy the given differential equation. It often includes an arbitrary constant or constants that account for different initial conditions or particular solutions. In the context of our given problem, the differential equation is \( \frac{dy}{dx} = \sec^2\left(\frac{x}{2}\right) \).
To find the general solution, we integrate the function \( \sec^2\left(\frac{x}{2}\right) \) with respect to \( x \). After integration, we get the expression \( y = 2\tan\left(\frac{x}{2}\right) + C \), where \( C \) is the constant of integration. This constant \( C \) allows the solution to represent not just a single curve, but an entire family of curves. This family of curves is the complete set of functions that satisfy the original differential equation. By varying \( C \), different particular solutions can be obtained.
In practice, if an initial condition is given, such as a point through which the solution must pass, \( C \) can be determined to find a particular solution. The general solution remains versatile and applicable across several scenarios, thus being a fundamental aspect of solving differential equations.

First-Order Differential Equation

A **first-order differential equation** involves derivatives of the function, but only up to the first derivative. It is generally in the form \( \frac{dy}{dx} = f(x, y) \). In our exercise, the equation presented is \( \frac{dy}{dx} = \sec^2\left(\frac{x}{2}\right) \).
This particular first-order equation is a bit simpler because it is already in a separated form, meaning we can directly integrate. First-order differential equations can often be solved using simple integration when the equation is separable, as in this case.
The key distinction in first-order equations is that they involve the dependent variable \( y \) and its first derivative \( \frac{dy}{dx} \). It's essential to recognize such structures because they dictate the methods of solution. Here, because \( y \) does not appear on the right-hand side after separating variables, we focus solely on integrating the expression involving \( x \).

• The solution provides the relationship between \( y \) and \( x \).
• The integration step usually involves simple calculus techniques.

Solving first-order differential equations is often an accessible entry point into understanding more complex differential equations.

Integration by Substitution

**Integration by substitution** is a method used to simplify the integration process, especially for functions that may appear challenging to integrate directly. It involves transforming the integral into a simpler form using a substitution variable.
In the given differential equation, we encountered the integral \( \int \sec^2\left(\frac{x}{2}\right) dx \). Direct integration could be complicated, but using substitution made the task easier.
Here's how we approached it:

• Set \( u = \frac{x}{2} \), then \( du = \frac{1}{2}dx \).
• Rewriting \( dx \) leads to \( dx = 2du \).

This substitution transformed our integral into \( \int \sec^2(u) 2 du \), which is more straightforward to solve because \( \int \sec^2(u) du = \tan(u) \). Performing the integration gives \( 2\tan(u) + C \).
Finally, substituting back for \( u \), we find \( u = \frac{x}{2} \), leading us back to \( 2\tan\left(\frac{x}{2}\right) + C \). Integration by substitution is a powerful tool because it allows us to reframe complex integrals into more manageable forms, aiding in the broader solution of differential equations.