Problem 36 - Calculus: Early Transcendentals

Question

Find a function \( f \) such that \( f(3) = 2 \) and \( (t^2 + 1)f'(t) + [f(t)]^2 + 1 = 0 t \neq 1 \)

Step-by-Step Solution

Answer

The function is \( f(t) = \tan(C - \arctan(t)) \) where \( C \) satisfies \( 2 = \tan(C - \arctan(3)) \).

Step 1: Understand the Given Conditions

We are asked to find a function \( f(t) \) that satisfies two conditions: \( f(3) = 2 \) and a differential equation \( (t^2 + 1)f'(t) + [f(t)]^2 + 1 = 0 \) for \( t eq 1 \). Our goal is to find such a function \( f(t) \).

Step 2: Analyze the Differential Equation

The differential equation is given by \( (t^2 + 1)f'(t) + [f(t)]^2 + 1 = 0 \). This equation implies that the derivative of \( f(t) \), \( f'(t) \), is needed. By isolating \( f'(t) \), we recognize it is a first-order nonlinear differential equation.

Step 3: Solve the Differential Equation

Re-write the differential equation to isolate \( f'(t) \) as follows: \( f'(t) = -\frac{[f(t)]^2 + 1}{t^2 + 1} \). Now separate variables to solve this: \( \int \frac{1}{[f(t)]^2 + 1} \, df = -\int \frac{1}{t^2 + 1} \, dt \).

Step 4: Integrate Both Sides

Perform the integration: The left side becomes \( \arctan(f(t)) + C_1 \), and the right side becomes \(-\arctan(t) + C_2\). Equating these gives \( \arctan(f(t)) = -\arctan(t) + C \) where \( C = C_2 - C_1 \).

Step 5: Solve for \( f(t) \)

From \( \arctan(f(t)) = -\arctan(t) + C \), solve for \( f(t) \) by applying the tangent function to both sides: \( f(t) = \tan(C - \arctan(t)) \). This is the general solution for \( f(t) \).

Step 6: Use Initial Condition to Find \( C \)

Apply the initial condition \( f(3) = 2 \) to find \( C \). Substitute \( t = 3 \) and \( f(3) = 2 \) into \( f(t) = \tan(C - \arctan(t)) \). Thus \( 2 = \tan(C - \arctan(3)) \). Compute \( \arctan(3) \) and solve for \( C \).

Step 7: Compute \( C \) and Write Final Function

Numerically evaluate or compute \( C \) from the equation \( 2 = \tan(C - \arctan(3)) \) using inverse trigonometric identities or numerical methods. Finally, write the function \( f(t) = \tan(C - \arctan(t)) \) with the determined \( C \).

Key Concepts

Nonlinear Differential Equation

In the exercise, we are dealing with a nonlinear differential equation, which means the equation involves powers, products, or other functions of the unknown function and its derivatives.
A nonlinear equation does not follow the principle of superposition, making them more challenging to solve than linear ones.
The given differential equation is:

• \((t^2 + 1)f'(t) + [f(t)]^2 + 1 = 0\)

Here, the function \( f(t) \) appears in a nonlinear term, \[ [f(t)]^2 \]. As such, traditional linear solving techniques won't work. Instead, separating variables—an integration technique—is needed to solve it.

Initial Conditions

Initial conditions provide specific values at certain points, helping to pinpoint the exact solution from multiple possibilities.
In our problem, the initial condition given is \( f(3) = 2 \). This condition ties down the general solution to a specific function.
Why are initial conditions important?

• They help solve for constants of integration that appear when integrating differential equations.
• They allow us to find a unique solution relevant to specific real-world problems.

In this case, after finding the general solution, we use \( f(3) = 2 \) to determine the constant \( C \). This process grounds the abstract solution, making it applicable within the set parameters.

Integration Techniques

Solving differential equations often requires integration. In this exercise, integration techniques help isolate and solve for the function \( f(t) \).
The process is as follows:

• First, separate variables: Rewrite the equation to express \( f'(t) \) on its own.
• Then, split the equation into two parts that can each be integrated separately.

The separated variables here are:

• \[ \int rac{1}{[f(t)]^2 + 1} \ df = -\int rac{1}{t^2 + 1} \ dt \]

To integrate these expressions:

• The left side becomes \( \arctan(f(t)) + C_1 \), using the formula for integrating the derivative of an arctan function.
• The right side becomes \(-\arctan(t) + C_2\).

Each side has their constants of integration, which can initially be denoted as \( C_1 \) and \( C_2 \). Finally, combine them with a single constant \( C = C_2 - C_1 \) to find the complete function later.

First-Order Differential Equations

Our problem involves a first-order differential equation, which means it contains the first derivative of the function \( f(t) \), with no higher derivatives.
First-order equations are fundamental in describing systems where the rate of change is a function of the state.
For the equation given in the exercise, the focus is on \

• Isolating \( f'(t) \) and tackling how \( f(t) \) changes with \( t \).

Key considerations when solving:

• Recognize and classify the equation type to apply suitable solving techniques, such as variable separation.
• Use integration to solve for the function.
• Apply initial conditions to find specific solutions.

Understanding first-order differential equations is crucial because they frequently model changes in diverse fields like physics, biology, and economics.