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------------------------------------------------ Find the particular solution for [tex]F(x)[/tex] if [tex]F'(x) = 62x^2 + 4x[/tex], and the initial condition is [tex]F(2) = 11[/tex].

Answer :

Final answer:

The particular solution for F(x) is F(x) = (62/3)x³ + 2x² - 10/3.

Explanation:

To find the particular solution for F(x) if F'(x) = 62x² + 4x, and the initial condition is F(2) = 11, we need to integrate the given derivative function. Using the power rule for integration, we integrate

62x² + 4x to get F(x) = (62/3)x³ + 2x² + C.

To determine the value of C, we use the initial condition F(2) = 11.

Plugging in x = 2 and F(x) = 11, we get the equation 11 = (62/3)(2³) + 2(2²) + C.

Simplifying this equation, we find C = -10/3. Therefore, the particular solution for F(x) is F(x) = (62/3)x³ + 2x² - 10/3.

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