Q. 43 Use antidifferentiation and/or s... [FREE SOLUTION]

Chapter 6: Q. 43 (page 570)

Use antidifferentiation and/or separation of variables to solve each of the initial-value problems in Exercises 29鈥�52
$\frac{d y}{d x} = 9.8 - \frac{0.3}{150} y, y (0) = 1000$

Short Answer

Expert verified

The solution of the initial-value problem $\frac{d y}{d x} = 9.8 - \frac{0.3}{150} y, y (0) = 1000 a s y (x) = 4900 - 3900 e^{\frac{- x}{500}}$

Step by step solution

Step 1. Given information

The given initial value problem $\frac{d y}{d x} = 9.8 - \frac{0.3}{150} y, y (0) = 1000 . . . . . . . . . . . (1)$

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

Note that the differential equation in (1) does not contain the independent variable at all, so technically the variables have already been separated. Hence, the differential equation can be solved by antidifferentiating. Thus, the solution of the differential equation involved in the initialvalue problem is given by

鈭� \frac{d y}{9.8 - \frac{0.3}{150} y} = 鈭� d x - \frac{150}{0.3} \ln | 9.8 - \frac{0.3}{150} y | = x + C_{1} \ln | 9.8 - \frac{0.3}{150} y | = - \frac{0.3}{150} x + C (- \frac{0.3}{150} C_{1} = C) 9.8 - \frac{0.3}{150} y = A e^{\frac{0.3}{150} x} (e^{c} = A)

Simplify the above expression further

\frac{0.3}{150} y = 9.8 - A e^{\frac{0.3}{150} x} y = 500 [9.8 - A e^{- \frac{1}{500} x}] = 4900 - 500 A e^{- \frac{1}{500} x}

Now, use the given initial condition

y (0) = 1000

, that is take

x = 0, y = 1000

in the above result and evaluate the constant A.

$1000 = 4900 - 500 A 500 A = 3900 A = \frac{39}{5}$

Substitute this value of the constant Ain the solution of the differential equation and obtain the solution of the initial-value problem $\frac{d y}{d x} = 9.8 - \frac{0.3}{150} y, y (0) = 1000 a s y (x) = 4900 - 3900 e^{\frac{- x}{500}}$

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 each of the initial-value problems in Exercises 29鈥�52
$\frac{d y}{d x} = 9.8 - \frac{0.3}{150} y, y (0) = 1000$

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

Pure Maths

Discrete Mathematics

Decision Maths

Logic and Functions

Calculus

Mechanics Maths

Study anywhere. Anytime. Across all devices.

91影视

Use antidifferentiation and/or separation of variables to solve each of the initial-value problems in Exercises 29鈥�52dydx=9.8-0.3150y,y(0)=1000

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

Pure Maths

Discrete Mathematics

Decision Maths

Logic and Functions

Calculus

Mechanics 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
$\frac{d y}{d x} = 9.8 - \frac{0.3}{150} y, y (0) = 1000$