Problem 23 - Advanced Engineering Mathematics

Question

From \(\frac{1}{x^{2}+1} d x=4 d t\) we obtain \(\tan ^{-1} x=4 t+c .\) Using \(x(\pi / 4)=1\) we find \(c=-3 \pi / 4 .\) The solution of the initial-value problem is \(\tan ^{-1} x=4 t-\frac{3 \pi}{4}\) or \(x=\tan \left(4 t-\frac{3 \pi}{4}\right)\).

Step-by-Step Solution

Answer

The solution is \(x = \tan(4t - \frac{3\pi}{4})\).

Step 1: Understanding the Differential Equation

We start with the differential equation given: \(\frac{1}{x^{2}+1} dx=4 dt\). This equation relates \(x\) and \(t\) and suggests a connection to inverse trigonometric functions.

Step 2: Identifying the Antiderivative

Recognize that the left side of the equation \(\frac{1}{x^{2}+1}\) is the derivative of \(\tan^{-1}x\). Therefore, integrating both sides gives \(\tan^{-1}x = 4t + C\), where \(C\) is the constant of integration.

Step 3: Applying Initial Conditions

We use the initial condition \(x(\pi/4) = 1\) to find \(C\). Substitute into the equation to get \(\tan^{-1}(1) = 4(\pi/4) + C\). Since \(\tan^{-1}(1) = \pi/4\), we solve for \(C\): \(\pi/4 = \pi + C\). Thus, \(C = -3\pi/4\).

Step 4: Writing the Particular Solution

Substitute \(C = -3\pi/4\) back into the equation \(\tan^{-1}x = 4t + C\). This gives us \(\tan^{-1}x = 4t - 3\pi/4\), which is the solution to the initial-value problem.

Step 5: Solving for \(x\)

To find \(x\), take the tangent of both sides of the equation: \(x = \tan(4t - 3\pi/4)\). This expresses \(x\) explicitly in terms of \(t\).

Key Concepts

Differential Equations

A differential equation is an equation that relates a function with its derivatives. It expresses a relationship involving rates of change and helps us model real-world dynamic systems such as physics problems or population growth. In the original problem, we have the equation \(\frac{1}{x^{2}+1} dx=4 dt\), which is a simple form of a differential equation. Here, we're looking for a function \(x(t)\) that describes how \(x\) changes as \(t\) changes.

Differential equations often require initial conditions, like \(x(\pi/4)=1\), to find specific solutions. These conditions allow us to calculate unknown constants that appear after integration and lead us to a unique, particular solution that fits the given scenario.

Inverse Trigonometric Functions

Inverse trigonometric functions are the reverse of the standard trigonometric functions like sine, cosine, and tangent. They allow us to find angles when we know the trigonometric ratios. In this exercise, the function \(\tan^{-1} x\) appears as a result of integrating \(\frac{1}{x^2+1}\), since \(\frac{d}{dx}[\tan^{-1} x] = \frac{1}{x^2+1}\).

This particular inverse function maps the tangent ratio back to an angle, which is crucial for solving problems in calculus where angles and arc lengths are involved. Understanding how to work with these inverse functions is important for reinterpreting the solutions of differential equations.

Integration

Integration is a fundamental concept in calculus that allows us to find the function from its derivative. It's essentially the inverse of differentiation. In the given problem, integration is used to solve the differential equation.

When we integrate \(\frac{1}{x^2+1}\), we find that its antiderivative is \(\tan^{-1} x\). This helps transform our differential equation into an expression involving \(\tan^{-1} x\) and \(t\). The process shows the role of constants of integration, denoted as \(C\), which must be determined using initial conditions to provide specific solutions.

Particular Solution

A particular solution is a solution to a differential equation that satisfies given initial conditions. Once we generalize a solution through integration, like \(\tan^{-1} x = 4t + C\), we use the initial condition \(x(\pi/4) = 1\) to find the specific constant \(C\).

By substituting in the initial conditions \(\tan^{-1}(1) = \pi/4\) and \(t = \pi/4\), we derive \(C = -3\pi/4\). This value gives us \(\tan^{-1} x = 4t - 3\pi/4\), our particular solution. Finally, we solve explicitly for \(x\), getting \(x = \tan(4t - 3\pi/4)\), which describes \(x\) in terms of \(t\) and fully answers the initial-value problem.