Q. 39 Use antidifferentiation and/or s... [FREE SOLUTION]

Chapter 6: Q. 39 (page 570)

Use antidifferentiation and/or separation of variables to solve each of the initial-value problems in Exercises 29鈥�52
$\frac{d y}{d x} = e^{x + y}, y (0) = 2$

Short Answer

Expert verified

The solution of the initial-value problem descibed by $\frac{d y}{d x} = e^{x + y}, y (0) = 2$ is $y (x) = - \ln | 1 + e^{- 2} - e^{x} |$

Step by step solution

Step 1. Given information

The given initial value problem $\frac{d y}{d x} = e^{x + y}, y (0) = 2 . . . . . . . . . . . . . . (1)$

Step 2. Use antidifferentiation and/or separation of variables to solve each of the initial-value

Use the property of exponential function and rewrite the differential equation in (1) as

\frac{d y}{d x} = e^{x} . e^{y} (鈭� e^{x} . e^{y} = e^{x + y})

Note that the differentlal equation is now in the form

\frac{d y}{d x} = p (x) q (y)

In which

p (x) = e^{x}

, and

q (y) = e^{y}

. So, the differential equation can be solved by applying variable separable method. Thus, the solution of the differential equation involved in the initial- value problem is given by

$鈭� \frac{1}{e^{y}} d y = 鈭� e^{x} d x 鈭� e^{- y} d y = e^{x} + C - e^{- y} = e^{x} + C e^{- y} = - e^{x} - C$

Now, use the given initial condition

y (0) = 2

, that is take

x = 0, y = 2

in the above result and evaluate the constant C.

$e^{- 2} = - 1 - C C = - e^{- 2} - 1$

Substitute this value of the constant Cin the solution of the differential equation and obtain the solution of the initial - value problem

\frac{d y}{d x} = e^{x + y}, y (0) = 2

$e^{- y} = - e^{x} + 1 + e^{- 2} - y = \ln | 1 + e^{- 2} - e^{x} | y = - \ln | 1 + e^{- 2} - e^{x} |$

Therefore,the solution of the initial-value problem descibed by equation(1) is

$y (x) = - \ln | 1 + e^{- 2} - e^{x} |$

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Use antidifferentiation and/or separation of variables to solve each of the initial-value problems in Exercises 29鈥�52
$\frac{d y}{d x} = e^{x + y}, y (0) = 2$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Use antidifferentiation and/or separation of variables to solve each of the initial-value

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91影视

Use antidifferentiation and/or separation of variables to solve each of the initial-value problems in Exercises 29鈥�52dydx=ex+y,y(0)=2

Short Answer

Step by step solution

Step 1. Given information

Step 2. Use antidifferentiation and/or separation of variables to solve each of the initial-value

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Mechanics Maths

Logic and Functions

Statistics

Discrete Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.

Use antidifferentiation and/or separation of variables to solve each of the initial-value problems in Exercises 29鈥�52
$\frac{d y}{d x} = e^{x + y}, y (0) = 2$