Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

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

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

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

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

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

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

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

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Calculus

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Short Answer

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

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

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,

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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,

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

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