Problem 25 - Calculus and its applications

Question

(a) find the particular solution of each differential equation as determined by the initial condition, and (b) check the solution by substituting into the differential equation. \(f^{\prime}(x)=x^{2 / 3}-x ; \quad f(1)=-6\)

Step-by-Step Solution

Answer

The particular solution is \( f(x) = \frac{3}{5}x^{5/3} - \frac{x^2}{2} - 6.1 \). It satisfies the given differential equation.

Step 1: Integrate the Differential Equation

To find the particular solution, we first need to integrate the differential equation. The given equation is \( f'(x) = x^{2/3} - x \). Integrate both sides to find \( f(x) \):\[ f(x) = \int (x^{2/3} - x)\, dx = \int x^{2/3} \, dx - \int x \, dx. \]Evaluate each integral separately:\[ \int x^{2/3} \, dx = \frac{x^{5/3}}{5/3} = \frac{3}{5}x^{5/3} \]\[ \int x \, dx = \frac{x^2}{2} \]. Thus, \( f(x) = \frac{3}{5}x^{5/3} - \frac{x^2}{2} + C \) where \( C \) is the integration constant.

Step 2: Apply the Initial Condition

The initial condition given is \( f(1) = -6 \). Use this to find \( C \):Substitute \( x = 1 \) and \( f(1) = -6 \) into the equation:\[ -6 = \frac{3}{5}(1)^{5/3} - \frac{(1)^2}{2} + C. \]Simplify this:\[ -6 = \frac{3}{5} \times 1 - \frac{1}{2} + C \]\[ -6 = \frac{3}{5} - \frac{5}{10} + C \]\[ -6 = \frac{3}{5} - \frac{1}{2} + C \]To combine \( \frac{3}{5} \) and \( \frac{1}{2} \), find a common denominator:\[ \frac{3}{5} = \frac{6}{10}, \quad \frac{1}{2} = \frac{5}{10} \]So\[ -6 = \frac{6}{10} - \frac{5}{10} + C = \frac{1}{10} + C \]Now solve for \( C \):\[ C = -6 - \frac{1}{10} = -6.1 \]. Thus, the particular solution is \( f(x) = \frac{3}{5}x^{5/3} - \frac{x^2}{2} - 6.1 \).

Step 3: Verify the Solution

Substitute the found particular solution back into the original differential equation:Differentiate \( f(x) = \frac{3}{5}x^{5/3} - \frac{x^2}{2} - 6.1 \) with respect to \( x \):\[ f'(x) = \frac{d}{dx} \left( \frac{3}{5}x^{5/3} - \frac{x^2}{2} - 6.1 \right) = \frac{3}{5} \times \frac{5}{3}x^{2/3} - x = x^{2/3} - x \].This matches the original differential equation \( f'(x) = x^{2/3} - x \). Therefore, the particular solution \( f(x) = \frac{3}{5}x^{5/3} - \frac{x^2}{2} - 6.1 \) is verified.

Key Concepts

Particular Solution

When dealing with differential equations, a particular solution is a solution that satisfies both the differential equation and any given conditions such as initial conditions. In the context of our exercise, the task was to find a specific function, achieved by integrating the derivative of the function and incorporating initial conditions. This solution is special in that it goes beyond a family of solutions, pinpointing exactly one solution that fits the conditions provided.
The process to find this solution starts by integrating the given equation to form the function whose derivative equals the given expression. Differentiating a function returns a derivative, but integrating gives us a whole family of functions. To narrow it down to one, as in our problem, initial conditions specify a unique solution by determining the correct integration constant. This leads us to the next core concept.

Initial Condition

Initial conditions are values provided to us when solving differential equations, and they help find a unique particular solution. These conditions often come as values specifying the function at a specific point. In this exercise, the initial condition is given as \( f(1) = -6 \).

• The role of an initial condition is to anchor the particular solution at a specific point in its domain.
• It determines the exact value of the function at a particular input value, guiding our integration process by providing enough data to find the unique solution.

Applying the initial condition involves substituting the specific value into the integrated solution, allowing us to solve for the integration constant. This ensures the solution curve passes through the specified initial point, fully aligning the mathematical solution with provided data.

Integration Constant

An integration constant appears when solving differential equations as a result of the indefinite integration process. This constant, often represented by \( C \), essentially represents the multiple solutions possible after integrating a differential equation. Each constant represents a shift of the entire function vertically.

• The role of the integration constant is particularly important because it signifies that the integration process alone isn't enough to pin down a single function without further conditions.
• In our problem, once we integrated the equation, \( f(x) = \frac{3}{5}x^{5/3} - \frac{x^2}{2} + C \) showed an arbitrary constant \( C \) which needed to be solved by employing the given initial condition.

By substituting \( f(1) = -6 \) into the fully integrated function and solving for \( C \), the constant was determined to be \( -6.1 \). This resolved the function to \( f(x) = \frac{3}{5}x^{5/3} - \frac{x^2}{2} - 6.1 \), completing our task to find the specific solution that not only satisfies the differential equation but also the initial condition.