Refer a friend and earn $50 when they subscribe to an annual planRefer Now

Get the answer to your homework problem.

Try Numerade Free for 30 Days

Like

Report

For the following problems, determine how parameter $a$ affects the solution.Solve the generic equation $y^{\prime}=a x+x y .$ How does varying $a$ change the behavior?

$y=-a+C e^{x^{2} / 2}$Hence a will have an effect if $C=0$ and $y$ will be a line parallel to $x$ axis which is $y=a$

Calculus 2 / BC

Chapter 4

Introduction to Differential Equations

Section 5

First-order Linear Equations

Differential Equations

University of Michigan - Ann Arbor

University of Nottingham

Idaho State University

Lectures

01:11

In mathematics, integratio…

06:55

In grammar, determiners ar…

04:01

For the following problems…

02:43

02:29

05:32

01:36

03:08

Use the variation-of-param…

06:59

Solve the equations in Exe…

12:24

01:25

Solve the equation $y ^ { …

Solve the equation $y^{\pr…

Section four, not five problem to 59. Here in this section, we're dealing with first order, linear, ordinary differential equations and solving these by finding and integrating factors. In this case, the key is to get this thing in standard form, which is why prime plus a function of X time is why equals another function off axe. So to get this one in standard form, that's why prime minus X y equal a X The integrating factor. It's just going to be e to the you take the anti derivative minus X dx that function p of X. So this gives us e to the minus. X squared over two is the integrating factor. So when I multiply the entire equation by that, I'll have you to the miners. Why e to the minus X squared over two Prime is equal to a X E to the minus x squared over to, and now the key will be to integrate both sides of this equation On the left side, we just get why e to the minus X squared over two on the right side of this equation. So the most straightforward way to do this Let's just scroll down. I need to integrate a x you to the miners X squared over two the X. So the most straightforward way of doing this is let you well, minus X squared over two. Then do you is going to be, um, Highness X DX so I can rewrite this one As this is equal to bring the A outside Put a minus there, minus x e to the minus, X squared over two t x. This is now just in the form of the integral of e of you, Do you, which is just e to the U plus a constant of integration. So what we end up with here is why e to the minus X squared over two is equal to minus a E to the minus X squared over two plus a constant of integration. Therefore, why is equal to minus a plus C e to the X squared over two. So that is my, um, answer to the, um uh, differential equation. And now we'll go take a look at the graph. And so here you see the graph of the solution for this different. Your equation y is equal to So why did you go to minus a C E to the X great over two. So when C is equal to zero, um, you're gonna have a horizontal line, So sorry. Let me change my mouse back. You're gonna have a horizontal line, A to minus a. So as a varies, you just see horizontal lines when CIA zero and then if c were allowed to vary based on a my initial condition, then that's where you see make this little bit clear. So if I scroll out, um, you see the exponential term, so you see the exponential term words increase decrease or a decrease increase based on the various values of A and C.

View More Answers From This Book

Find Another Textbook

Numerade Educator

In mathematics, integration is one of the two main operations in calculus, w…

In grammar, determiners are a class of words that are used in front of nouns…

For the following problems, determine how parameter $a$ affects the solution…

For the following problems, find the general solution to the differential eq…

Use the variation-of-parameters method to solve the given differential equat…

Solve the equations in Exercises by variation of parameters.$$y^{\prime …

Use the variation-of-parameters method to find the general solution to the g…

Solve the equation $y ^ { \prime } = x \sqrt { x ^ { 2 } + 1 } / \left( y e …

Solve the equation $y^{\prime}=x \sqrt{x^{2}+1} /\left(y e^{y}\right)$ and g…

01:42

Determine whether the sequence defined as follows has a limit. If it does, f…

01:34

Find the volume of the solid generated by revolving the region in the first …

02:45

Show that the population grows fastest when it reaches half the carrying cap…

01:02

Use the comparison test to determine whether the following series converge. …

02:26

For each pair of integrals, determine which one is more difficult to evaluat…

Using sigma notation, write the following expressions as infinite series.

05:28

For the following exercises, integrate using whatever method you choose.…

01:41

The following problems consider the logistic equation with an added term for…

00:25

Does $\sum_{n=1}^{\infty} 2^{-\ln \ln n}$ converge? (Hint: Write $2^{\ln \ln…

00:58

What are the integrating factors for the following differential equations?

92% of Numerade students report better grades.

Try Numerade Free for 30 Days. You can cancel at any time.

Annual

0.00/mo 0.00/mo

Billed annually at 0.00/yr after free trial

Monthly

0.00/mo

Billed monthly at 0.00/mo after free trial

Earn better grades with our study tools:

Textbooks

Video lessons matched directly to the problems in your textbooks.

Ask a Question

Can't find a question? Ask our 30,000+ educators for help.

Courses

Watch full-length courses, covering key principles and concepts.

AI Tutor

Receive weekly guidance from the world’s first A.I. Tutor, Ace.

30 day free trial, then pay 0.00/month

30 day free trial, then pay 0.00/year

You can cancel anytime

OR PAY WITH

Your subscription has started!

The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

Receive weekly guidance from the world's first A.I. Tutor, Ace.

Mount Everest weighs an estimated 357 trillion pounds

Snapshot a problem with the Numerade app, and we'll give you the video solution.

A cheetah can run up to 76 miles per hour, and can go from 0 to 60 miles per hour in less than three seconds.

Back in a jiffy? You'd better be fast! A "jiffy" is an actual length of time, equal to about 1/100th of a second.