Problem 106 - Thomas Calculus in SI Units

Question

Solve the initial value problems. $$\frac{d y}{d x}=\frac{1}{1+x^{2}}-\frac{2}{\sqrt{1-x^{2}}}, \quad y(0)=2$$

Step-by-Step Solution

Answer

The solution is \( y = \arctan(x) - 2\arcsin(x) + 2 \).

Step 1: Identify the Differential Equation

The given differential equation is \( \frac{dy}{dx} = \frac{1}{1+x^2} - \frac{2}{\sqrt{1-x^2}} \) with the initial condition \( y(0) = 2 \). This is a first-order differential equation.

Step 2: Separate the Integrals

The expression on the right-hand side is composed of two separate functions: \( \frac{1}{1+x^2} \) and \( -\frac{2}{\sqrt{1-x^2}} \). We will handle each part independently through integration.

Step 3: Integrate the First Term

Integrate \( \frac{1}{1+x^2} \) with respect to \( x \). The antiderivative of \( \frac{1}{1+x^2} \) is \( \arctan(x) \), so the integral is \( \int \frac{1}{1+x^2} \, dx = \arctan(x) + C_1 \).

Step 4: Integrate the Second Term

For the second term, integrate \( -\frac{2}{\sqrt{1-x^2}} \) with respect to \( x \). The antiderivative of \( \frac{1}{\sqrt{1-x^2}} \) is \( \arcsin(x) \), hence the integral is \( \int -\frac{2}{\sqrt{1-x^2}} \, dx = -2\arcsin(x) + C_2 \).

Step 5: Combine the Integrals and Form the Solution

Combine the results from Steps 3 and 4: \( y = \arctan(x) - 2\arcsin(x) + C \).

Step 6: Apply the Initial Condition

Use the initial condition \( y(0) = 2 \) to find \( C \). Substitute \( x = 0 \) and \( y = 2 \) into the equation: \( 2 = \arctan(0) - 2\arcsin(0) + C \). This simplifies to \( 2 = 0 - 0 + C \), so \( C = 2 \).

Step 7: Write the Final Solution

Incorporating the value of \( C \), the solution to the initial value problem is: \( y = \arctan(x) - 2\arcsin(x) + 2 \).

Key Concepts

Initial Value Problems

Initial value problems in differential equations are scenarios where we are given a differential equation, along with an initial condition. This condition initially specifies the value of the unknown function at a certain point. For example, the problem gives us

• a differential equation: \( \frac{dy}{dx} = \frac{1}{1+x^2} - \frac{2}{\sqrt{1-x^2}} \)
• an initial condition: \( y(0) = 2 \).

The goal is to find the specific solution that not only satisfies the differential equation but also meets this initial condition. Initial value problems are crucial as they model systems that evolve from a known starting point, making them very applicable to real-world situations like population dynamics and electrical circuits.
Recognizing that we have an initial condition allows us to determine any additional constants of integration, ensuring our solution is unique.

First-Order Differential Equation

A first-order differential equation is characterized by the highest derivative being the first derivative. In our problem, the given equation \( \frac{dy}{dx} = \frac{1}{1+x^2} - \frac{2}{\sqrt{1-x^2}} \) involves only the first derivative of \( y \) with respect to \( x \).
First-order differential equations have one important aspect - they describe rate processes, such as velocity or growth rates, which are fundamental in many scientific fields.

• They can be of multiple forms, and understanding these forms helps in identifying the suitable method to solve them.
• Our equation is linear in terms of the function \( y \), which typically makes the integration straightforward.

Being able to recognize and categorize the differential equation is the first step towards solving it efficiently.

Integration Techniques

Integration techniques allow us to find the antiderivative of functions. In the context of differential equations, particularly first-order ones, these techniques are invaluable for finding solutions.
For our given problem, we split the equation into manageable parts:

• \( \int \frac{1}{1+x^2} \, dx = \arctan(x) + C_1 \)
• \( \int -\frac{2}{\sqrt{1-x^2}} \, dx = -2\arcsin(x) + C_2 \)

Both of these are standard integrals, with the first representing the integration rule for the inverse tangent and the latter for the inverse sine. Recognizing these patterns is key to efficiently solving problems without getting bogged down in complex algebra. It is through mastering these techniques that we can compose solutions to differential equations.

Solution of Differential Equations

After integrating the separate parts of the equation, the final step is to combine them into a full solution. For our initial value problem, the solution is expressed as \( y = \arctan(x) - 2\arcsin(x) + C \).
To incorporate the initial condition \( y(0) = 2 \), we calculate the constant \( C \). Plugging \( x = 0 \) into the integrated equation, we find:

• \( 2 = \arctan(0) - 2\arcsin(0) + C \)
• Simplifying yields \( 2 = 0 - 0 + C \), so \( C = 2 \)

Thus, the specific solution that satisfies both the differential equation and the initial condition is \( y = \arctan(x) - 2\arcsin(x) + 2 \). This process of solving a differential equation involves not just mathematical computations, but also applying logical reasoning to ensure completeness and correctness of the solution. Understanding each part of the solution path enhances one's ability to tackle different types of differential equations in future problems.