/*! 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 21 Find the general solution of the... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 21

Find the general solution of the following equations. $$y^{\prime}(t)=3 y-4$$

Short Answer

Expert verified
Answer: The general solution of the given ODE is $y(t) = \frac{4}{3} + Ce^{3t}$.

Step by step solution

01

Identify the differential equation type

Determine whether the given differential equation is linear or not. In this case, the given ODE is a first-order linear ODE: $$y^{\prime}(t)=3 y-4$$
02

Write the equation in standard form

Write the given ODE in its standard linear form: $$y^{\prime}(t) -3y = -4$$
03

Find the integrating factor

Calculate the integrating factor: $$\mu(t) = e^{\int{-3\, dt}} = e^{-3t}$$
04

Multiply the differential equation by the integrating factor

Multiply the standard form ODE by the integrating factor: $$e^{-3t}(y^{\prime}(t) -3y) = e^{-3t}(-4)$$
05

Check that the left-hand side is an exact derivative

Check if the left-hand side is an exact derivative, and in this case, it is: $$\frac{d}{dt}(e^{-3t}y(t)) = e^{-3t}(y^{\prime}(t) -3y)$$
06

Integrate both sides with respect to t

Integrate both sides of the equation with respect to t: $$\int \frac{d}{dt}(e^{-3t}y(t)) \, dt = \int e^{-3t}(-4) \, dt$$
07

Solve the integral

Solving the integral, we get: $$e^{-3t}y(t) = \frac{4}{3}e^{-3t} + C$$
08

Isolate y(t) to find the general solution

Finally, isolate y(t) to find the general solution of the ODE: $$y(t) = \frac{4}{3} + Ce^{3t}$$

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

A differential equation of the form \(y^{\prime}(t)=F(y)\) is said to be autonomous (the function \(F\) depends only on \(y\) ). The constant function \(y=y_{0}\) is an equilibrium solution of the equation provided \(F\left(y_{0}\right)=0\) (because then \(y^{\prime}(t)=0,\) and the solution remains constant for all \(t\) ). Note that equilibrium solutions correspond to horizontal line segments in the direction field. Note also that for autonomous equations, the direction field is independent of \(t\). Consider the following equations. a. Find all equilibrium solutions. b. Sketch the direction field on either side of the equilibrium solutions for \(t \geq 0\). c. Sketch the solution curve that corresponds to the initial condition \(y(0)=1\). $$y^{\prime}(t)=y(y-3)$$

Apply Simpson's Rule to the following integrals. It is easiest to obtain the Simpson's Rule approximations from the Trapezoid Rule approximations, as in Example \(7 .\) Make \(a\) table similar to Table 7.8 showing the approximations and errors for \(n=4,8,16,\) and \(32 .\) The exact values of the integrals are given for computing the error. \(\int_{1}^{e} \ln x d x=1\)

A remarkable integral It is a fact that \(\int_{0}^{\pi / 2} \frac{d x}{1+\tan ^{m} x}=\frac{\pi}{4}\) for all real numbers \(m .\) a. Graph the integrand for \(m=-2,-3 / 2,-1,-1 / 2,0,1 / 2\) \(1,3 / 2,\) and \(2,\) and explain geometrically how the area under the curve on the interval \([0, \pi / 2]\) remains constant as \(m\) varies. b. Use a computer algebra system to confirm that the integral is constant for all \(m.\)

An important function in statistics is the Gaussian (or normal distribution, or bell-shaped curve), \(f(x)=e^{-\alpha x^{2}}\). a. Graph the Gaussian for \(a=0.5,1,\) and 2 b. Given that \(\int_{-\infty}^{\infty} e^{-a x^{2}} d x=\sqrt{\frac{\pi}{a}},\) compute the area under the curves in part (a). c. Complete the square to evaluate \(\int_{-\infty}^{\infty} e^{-\left(a x^{2}+b x+c\right)} d x,\) where \(a>0, b,\) and \(c\) are real numbers.

Use integration by parts to evaluate the following integrals. $$\int_{0}^{\infty} x e^{-x} d x$$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

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.