Problem 74 - University Calculus: Early Transcendentals

Question

Solve the initial value problems $$\frac{d y}{d x}=4 x\left(x^{2}+8\right)^{-1 / 3}, \quad y(0)=0$$

Step-by-Step Solution

Answer

The solution is \( y = 3(x^2+8)^{2/3} - 12 \).

Step 1: Rewriting the Differential Equation

Given the differential equation \( \frac{d y}{d x} = 4x(x^2+8)^{-1/3} \), first identify that it is a separable differential equation. Here, \( rac{d y}{d x} \) is written in terms of \( x \).

Step 2: Separating Variables

Rearrange the equation to separate variables: \( dy = 4x(x^2+8)^{-1/3} dx \). This expresses \( dy \) in terms of \( x \) and \( dx \), preparing us for integration.

Step 3: Integrating Both Sides

Integrate both sides of the equation: \( \int dy = \int 4x(x^2+8)^{-1/3} dx \). The left side becomes \( y \), and we'll focus on integrating the right side.

Step 4: Substituting for Integration

Use substitution to solve the integral on the right hand side. Let \( u = x^2 + 8 \) which means \( du = 2x \, dx \). Then, \( x \, dx = \frac{1}{2} du \). Substitute to get: \( \int 4x(x^2+8)^{-1/3} dx = \int 2u^{-1/3} du \).

Step 5: Integrating with Respect to \( u \)

Integrate the simplified equation: \( \int 2u^{-1/3} \, du \). The result is \( 2 \cdot \frac{u^{2/3}}{2/3} = 3u^{2/3} + C \).

Step 6: Substitute back \( u(x) \)

Replace \( u \) back with \( x^2+8 \): \( y = 3(x^2+8)^{2/3} + C \).

Step 7: Applying the Initial Condition

Use the initial condition \( y(0) = 0 \). Substitute \( x = 0 \) and \( y = 0 \) into the equation: \( 0 = 3(0^2 + 8)^{2/3} + C \). Solve for \( C \): \( 0 = 3(8)^{2/3} + C \) gives \( C = -3 \cdot 4 = -12 \).

Step 8: Final Solution

Substitute \( C = -12 \) back into the equation for \( y \). The solution becomes \( y = 3(x^2+8)^{2/3} - 12 \).

Key Concepts

Initial Value Problems

Initial value problems are a crucial part of differential equations. They help us find a specific solution that fits a designated condition. When you solve a differential equation, you often find a general solution that includes an arbitrary constant. An initial value problem (IVP), however, requires you to determine this constant so that the solution meets given initial conditions.

In the problem at hand, y(0) = 0 is our initial condition. This tells us that when x = 0, y equals 0. By substituting these values into the general solution, we pinpoint the constant C. This way, we tailor our equation's solution to pass through the precise point needed.

Working with initial value problems means you're not just painting the whole picture of possible solutions, but focusing on one that starts at a known point. This makes your solution applicable and meaningful in real scenarios.

Integration by Substitution

Integration by substitution is akin to unpicking a complex knot by replacing it with something simpler. It's incredibly useful when dealing with differential equations needing complex integration.

The substitution technique replaces a complicated part of the equation with a single variable. In our example, we used the substitution method by letting \( u = x^2 + 8 \). This change effectively redefined our integral expression, making it much easier to solve.

• Set \( u = x^2 + 8 \)
• Calculate \( du = 2x \, dx \)
• Reformulate the integral as \( \int 2u^{-1/3} \, du \)

Using this substitution, the complexity of the original integration task is reduced. Once the new integral is solved, you reverse the substitution process to get back to the terms of your original variables. This step is crucial for obtaining a solution that answers the posed problem.

Differential Equations Solution Steps

Solving differential equations systematically with clear steps ensures accuracy and understanding. Let's break down this process into digestible parts:

• **Identify and Separate Variables:** Start with the differential equation \( \frac{d y}{d x} = 4x(x^2+8)^{-1/3} \). Recognize it as a separable equation and maneuver it so that terms involving \( y \) are on one side and terms involving \( x \), along with \( dx \), are on the other side.
• **Integrate:** Once separated, integrate both sides. The left side directly becomes \( y \). The right side requires thoughtful integration techniques—like substitution in this exercise.
• **Apply Substitution:** Simplify the integration process on the right side by using substitution, a useful strategy when dealing with non-standard forms.
• **Solve for the Integration Constant:** After finding the integral's generic solution, use the initial condition provided to solve for the constant \( C \).
• **Representation of the Final Solution:** Finally, substitute back any interim variables and the constant to present the overall solution to the differential equation.

These steps ensure that you're not missing critical parts in problem-solving and guide you from the start to the meaningful outcome of your initial value problem.