Q. 19 Use antidifferentiation and/or s... [FREE SOLUTION]

Chapter 6: Q. 19 (page 570)

Use antidifferentiation and/or separation of variables to solve the given differential equations. Your answers will involve unsolved constants.
$\frac{d y}{d x} = {(x^{3} + 4)}^{2}$

Short Answer

Expert verified

Ans: The solution of the differential equation $\frac{d y}{d x} = {(x^{3} + 4)}^{2}$ is $\frac{1}{7} x^{7} + 2 x^{4} + 16 x + C$ .

Step by step solution

Step 1. Given information.

given,

$\frac{d y}{d x} = {(x^{3} + 4)}^{2}$

Step 2. Consider the differential equation defined by equation (1) given below and solve it by using antidifferentiation and/or separation of the variable method.

$\frac{d y}{d x} = {(x^{3} + 4)}^{2} . . . . . (1)$

Step 3. Now,

Note that the differential equation $(1)$ does not contain the dependent variable at all, so technically the variables have already been separated. So, the differential equation can be solved by antidifferentiation. Thus, the solution of the differential equation is obtained by integrating both the sides

$\begin{aligned} 鈭� d y = 鈭� {(x^{3} + 4)}^{2} d x \\ = 鈭� (x^{6} + 8 x^{3} + 16) d x \\ = 鈭� x^{6} d x + 8 鈭� x^{3} d x + 16 鈭� d x \\ = \frac{1}{7} x^{7} + 2 x^{4} + 16 x + C \end{aligned}$

Hence, a solution to the differential equation role="math" localid="1649151183048" $\frac{d y}{d x} = {(x^{3} + 4)}^{2}$ is $\frac{1}{7} x^{7} + 2 x^{4} + 16 x + C .$

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

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Use antidifferentiation and/or separation of variables to solve the given differential equations. Your answers will involve unsolved constants.
$\frac{d y}{d x} = {(x^{3} + 4)}^{2}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Consider the differential equation defined by equation (1) given below and solve it by using antidifferentiation and/or separation of the variable method.

Step 3. Now,

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Logic and Functions

Discrete Mathematics

Geometry

Calculus

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

91影视

Use antidifferentiation and/or separation of variables to solve the given differential equations. Your answers will involve unsolved constants. dydx=x3+42

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Consider the differential equation defined by equation (1) given below and solve it by using antidifferentiation and/or separation of the variable method.

Step 3. Now,

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Logic and Functions

Discrete Mathematics

Geometry

Calculus

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Use antidifferentiation and/or separation of variables to solve the given differential equations. Your answers will involve unsolved constants.
$\frac{d y}{d x} = {(x^{3} + 4)}^{2}$