Problem 12 - The Calculus with Analytic Geometry

Question

\(\frac{d y}{d x}=\frac{x}{4 \sqrt{\left(1+x^{2}\right)^{3}}} ; y=0\) when \(x=1\)

Step-by-Step Solution

Answer

The solution is \[ y = \frac{1}{8}(1+x^2)^{3/2} - \frac{1}{4} \].

Step 1: Identify the differential equation

The given differential equation is \( \frac{dy}{dx} = \frac{x}{4 \sqrt{(1+x^2)^3}} \).

Step 2: Separate the variables

Rewrite the differential equation to separate variables: \( 4 \sqrt{(1+x^2)^3} \, dy = x \, dx \).

Step 3: Integrate both sides

Integrate both sides of the equation: \[ 4 \int \sqrt{(1+x^2)^3} \, dy = \int x \, dx \]. The integrals are \( 4y \) on the left and \( \frac{1}{2}(1+x^2)^{3/2} + C_1 \) on the right.

Step 4: Combine the constants

Combine your constants of integration: \[ 4y = \frac{1}{2}(1+x^2)^{3/2} + C_1 \].

Step 5: Solve for the constant using initial conditions

Substitute the initial condition \( y=0 \) when \( x=1 \) to find \( C_1 \). \[ 4(0) = \frac{1}{2}(1+1^2)^{3/2} + C_1 \], solving gives \( C_1 = -1 \).

Step 6: Write the particular solution

Replace \( C_1 \) in the combined integral to find the particular solution: \[ y = \frac{1}{8}(1+x^2)^{3/2} - \frac{1}{4} \].

Key Concepts

Differential Equations

A differential equation is a mathematical equation that relates a function with its derivatives. In simpler terms, it describes how a particular quantity changes with respect to another. For this exercise, we have the differential equation: \( \frac{d y}{d x} = \frac{x}{4 \sqrt{(1+x^2)^3}} \).
Differential equations are fundamental in describing various phenomena in science and engineering, such as motion, electrical circuits, and growth processes. They allow us to predict how a system evolves over time. Differential equations can be ordinary (ODE) or partial (PDE) depending on whether they involve derivatives with respect to one or more variables. In our case, we have an ordinary differential equation (ODE) involving the variables x and y.

Integration

Integration is the process of finding the function F whose derivative is a given function f. This process is crucial for solving differential equations. In our exercise, after separating the variables, we need to integrate:

• The left side which involves the variable y: \( 4 \int \sqrt{(1+x^2)^3} \, dy \)
• The right side which involves the variable x: \( \int x \cdot dx \)

Integration requires techniques such as substitution, integration by parts, or using tables of integrals. Here, the integral on the left simplifies directly to 4y and the integral on the right results in \( \frac{1}{2}(1+x^2)^{3/2} + C_1 \). These steps are essential for moving forward to solve for the particular solution.

Initial Conditions

Initial conditions are values specified for the solution at a particular point, which help determine the unique solution to a differential equation. In our exercise, we have the initial condition \( y=0 \) when \( x=1 \)

Initial conditions help us find the integration constant C. By substituting the initial condition into the integrated solution, we solve for C. For instance, substituting \( y=0 \) and \( x=1 \) into the integrated equation gives us:
\[ 4(0) = \frac{1}{2}(1+1^2)^{3/2} + C_1 \]
Solving this equation yields \( C_1 = -1 \). Using initial conditions ensures that the solution fits the specific situation described by the differential equation.

Particular Solution

The particular solution to a differential equation is a specific solution that satisfies the differential equation and any given initial or boundary conditions. Once we determine the value of the integration constant using the initial conditions, we substitute it back into the integrated equation.
In our example, substituting \( C_1 = -1 \) back into the equation gives us:
\[ 4y = \frac{1}{2}(1+x^2)^{3/2} - 1 \]
Solving for y, we get the particular solution:
\[ y = \frac{1}{8}(1+x^2)^{3/2} - \frac{1}{4} \]
This particular solution satisfies both the differential equation and the initial condition, providing a comprehensive answer to the problem.