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Chapter 4: Problem 69

Find the solution of the following initial value problems. $$g^{\prime}(x)=7 x\left(x^{6}-\frac{1}{7}\right) ; g(1)=2$$

Short Answer

Expert verified
Answer: The solution is $$g(x) = \frac{7x^8}{8} - \frac{x^2}{2} + \frac{13}{8}.$$

Step by step solution

01

Identify the given terms and initial value

We are given the derivative of the function g, which is: $$g^{\prime}(x)=7 x\left(x^{6}-\frac{1}{7}\right)$$ And the initial value when x = 1: $$g(1)=2$$
02

Integrate the derivative

To find the function g(x), we need to integrate the given derivative with respect to x: $$g(x) = \int 7x\left(x^6 - \frac{1}{7}\right)dx$$
03

Simplify the integrand

We can distribute the term '7x' inside the parentheses to simplify the expression before integrating: $$g(x) = \int \left(7x^7 - x\right)dx$$
04

Perform the integration

Now we integrate the simplified expression with respect to x: $$g(x) = \int (7x^7 - x)dx = \frac{7x^8}{8} - \frac{x^2}{2} + C$$ Here, C is the constant of integration.
05

Use the initial condition to find the constant of integration

Since we are given that g(1) = 2, we can plug in x = 1 into g(x) and solve for C: $$2 = \frac{7(1)^8}{8} - \frac{(1)^2}{2} + C$$ $$2 = \frac{7}{8} - \frac{1}{2} + C$$ Solve for C: $$C = 2 - \frac{7}{8} + \frac{4}{8} = 2 - \frac{3}{8}$$ $$C = \frac{13}{8}$$
06

Write the final solution

The solution to the given initial value problem is the function g with the found constant of integration: $$g(x) = \frac{7x^8}{8} - \frac{x^2}{2} + \frac{13}{8}$$

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