/*! 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 2 Solve the differential equation.... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 2

Solve the differential equation. $$ \frac{d y}{d x}=4-y $$

Short Answer

Expert verified
The solution to the given differential equation is \(y = 4 卤 C1e^{-x}\).

Step by step solution

01

Rearrange The Equation

Rearrange the differential equation in order to separate the variables. This is done by moving all terms involving \(y\) to one side and all terms involving \(x\) to the other.\nSo, we re-write the differential equation as, \(dy/(4 - y) = dx.\)
02

Integrate Both Sides

Once the equation is rearranged, the next step is to integrate both sides. When integrating, remember to include the constant of integration 'C'.\nThus, we obtain, \(\int dy/(4 - y) = \int dx.\) This will result in \(- ln|4-y| = x + C. \)
03

Solve For Y

To isolate \(y\), first, remove the natural logarithm using an exponential function. This will give |4-y| = e^(-x-C). Then, write this absolute value equation as a piecewise function and solve for \(y\).\nSo, finally we get, \(y = 4 - e^{C}e^{-x}\) or \(y = 4 + e^{C}e^{-x}\). Note that \(e^{C}\) is just another constant, we can denote it as \(C1\). Therefore, the general solution to the given differential equation is \(y = 4 卤 C1e^{-x}\).

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

The region bounded by \(y=\sqrt{x}, y=0, x=0,\) and \(x=4\) is revolved about the \(x\) -axis. (a) Find the value of \(x\) in the interval [0,4] that divides the solid into two parts of equal volume. (b) Find the values of \(x\) in the interval [0,4] that divide the solid into three parts of equal volume.

Use the disk method to verify that the volume of a sphere is \(\frac{4}{3} \pi r^{3}\).

Let \(V\) be the region in the cartesian plane consisting of all points \((x, y)\) satisfying the simultaneous conditions \(|x| \leq y \leq|x|+3\) and \(y \leq 4\) Find the centroid \((\bar{x}, \bar{y})\) of \(V\).

A tank on a water tower is a sphere of radius 50 feet. Determine the depths of the water when the tank is filled to one-fourth and three-fourths of its total capacity. (Note: Use the zero or root feature of a graphing utility after evaluating the definite integral.)

Sketch the region bounded by the graphs of the algebraic functions and find the area of the region. $$ y=\frac{1}{x^{2}}, \quad y=0, \quad x=1, \quad x=5 $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Geometry

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.