/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 69 Find the solution of the followi... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

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 particular solution to the initial value problem is \(g(x) = \frac{7}{8}x^8 - \frac{1}{2}x^2 + \frac{13}{8}\).

Step by step solution

01

Find the antiderivative of function \(g'(x)\)

We want to find the antiderivative of the function \(g'(x) = 7x(x^6 - \frac{1}{7})\). To integrate this function, we can first distribute the 7x through the parentheses, which gives $$g'(x) = 7x^7 - x.$$ Now, we can find the antiderivative of this function: $$g(x) = \int g'(x) dx = \int (7x^7 - x) dx.$$ We integrate term-by-term to find the antiderivative: $$g(x) = \frac{7}{8}x^8 - \frac{1}{2}x^2 + C,$$ where C is the constant of integration.
02

Use the initial condition to find the constant of integration

The initial condition is \(g(1) = 2\). We can substitute this into the function we found in step 1, which is: $$g(1) = \frac{7}{8}(1)^8 - \frac{1}{2}(1)^2 + C = 2.$$ This simplifies to $$C = 2 - \left(\frac{7}{8} - \frac{1}{2}\right)$$ $$C = 2 - \frac{3}{8} = \frac{13}{8}.$$
03

Write the particular solution of the IVP

Now that we have found the constant of integration, we can write the particular solution to the initial value problem: $$g(x) = \frac{7}{8}x^8 - \frac{1}{2}x^2 + \frac{13}{8}.$$ This function satisfies the given condition \(g(1) = 2\) and is the solution to the initial value problem.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Determine the following indefinite integrals. Check your work by differentiation. $$\int \frac{2+x^{2}}{1+x^{2}} d x$$

Sketch the graph of a function that is continuous on \((-\infty, \infty)\) and satisfies the following sets of conditions. $$\begin{array}{l}f^{\prime \prime}(x)>0 \text { on }(-\infty,-2) ; f^{\prime \prime}(x)<0 \text { on }(-2,1) ; f^{\prime \prime}(x)>0 \text { on } \\\\(1,3) ; f^{\prime \prime}(x)<0 \text { on }(3, \infty)\end{array}$$

Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position. $$a(t)=2 \cos t ; v(0)=1, s(0)=0$$

Determine the following indefinite integrals. Check your work by differentiation. $$\int(\sqrt[3]{x^{2}}+\sqrt{x^{3}}) d x$$

Use analytical methods to evaluate the following limits. $$\lim _{x \rightarrow 1^{+}}\left(\frac{1}{x-1}-\frac{1}{\sqrt{x-1}}\right)$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.