Questions: Find the particular solution that satisfies the differential equation and the initial condition. f'(x) = 1/8 x - 4 ; f(16) = -48 f(x) =

Find the particular solution that satisfies the differential equation and the initial condition. f'(x) = 1/8 x - 4 ; f(16) = -48 f(x) =
Transcript text: Find the particular solution that satisfies the differential equation and the initial condition. \[ \begin{array}{l} \quad f^{\prime}(x)=\frac{1}{8} x-4 ; \quad f(16)=-48 \\ f(x)=\square \end{array} \]
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Solution

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Solution Steps

To find the particular solution that satisfies the given differential equation and initial condition, we need to integrate the differential equation to find the general solution and then use the initial condition to find the particular solution.

  1. Integrate the differential equation \( f'(x) = \frac{1}{8}x - 4 \) to find the general solution \( f(x) \).
  2. Use the initial condition \( f(16) = -48 \) to solve for the constant of integration.
Step 1: Integrate the Differential Equation

Given the differential equation: \[ f^{\prime}(x) = \frac{1}{8}x - 4 \]

To find \( f(x) \), we need to integrate \( f^{\prime}(x) \).

\[ f(x) = \int \left( \frac{1}{8}x - 4 \right) \, dx \]

Step 2: Perform the Integration

Integrate each term separately:

\[ \int \frac{1}{8}x \, dx = \frac{1}{8} \int x \, dx = \frac{1}{8} \cdot \frac{x^2}{2} = \frac{1}{16}x^2 \]

\[ \int -4 \, dx = -4x \]

Combining these results, we get:

\[ f(x) = \frac{1}{16}x^2 - 4x + C \]

where \( C \) is the constant of integration.

Step 3: Apply the Initial Condition

We are given the initial condition \( f(16) = -48 \). Substitute \( x = 16 \) and \( f(16) = -48 \) into the equation to solve for \( C \):

\[ -48 = \frac{1}{16}(16)^2 - 4(16) + C \]

Simplify the equation:

\[ -48 = \frac{1}{16} \cdot 256 - 64 + C \]

\[ -48 = 16 - 64 + C \]

\[ -48 = -48 + C \]

\[ C = 0 \]

Final Answer

Substitute \( C = 0 \) back into the equation for \( f(x) \):

\[ f(x) = \frac{1}{16}x^2 - 4x \]

Thus, the particular solution is:

\[ \boxed{f(x) = \frac{1}{16}x^2 - 4x} \]

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