Find the general solution of the following differential equations (complementary function particular solution). Find the solution by inspection or by (6.18), (6.23), or (6.24). Also find a computer solution and reconcile differences if necessary, noticing especially whether the solution is in simplest form [see (6.26) and the discussion after (6.15)].

(D2+1)y=2ex

Short Answer

Expert verified

The general solution of the differential equation is.y=yc+yp=α1eix+α2eix+ex

Step by step solution

01

Given information 

A differential equation is given as.(D22D+1)y=0

02

Auxiliary equation 

-Auxiliary equation:

Auxiliary equation is an algebraic equation of degreenupon which depends the solution of a given nth-order differential equation or difference equation.

-General form of the auxiliary equation (Da)(Db)=kecx

03

Roots of the auxiliary equation

By guessing the particular- solution ypfor this differential equation, which is yp=exand it is easy to prove that this is really a solution for this differential equation. Now to find the particular- solution for this differential equation by solving it have zero right-hand side, that is

(D2+1)y=0(D+i)(Di)y=0

Notice that, the roots of the auxiliary equation are complex, and not equal, so the solution would be in the form of eq. (5.l6). Now, any solution for the equation (D+i)y=0is also a solution to our differential equation (see the textbook page 409 for more details).

y'+iy=0dydx=iydyy=idxy=α1eix

04

General solution of differential equation 

By doing the same thing for the other simple equation, (Di)y=0and get

y=α2eix

So, the complementary solution is

yc=α1eix+α2eix

And therefore, the general solution of the differential equation(D2+1)y=0 is

y=yc+yp=α1eix+α2eix+ex

By computer software write Simplify [DSolve[y,,[x]+y[x]==2Ex,y[x],x]]will get

y=ex+c1cosx+c2sinx

But rewrite this result (i.e., the complementary solution) if write the exponential in terms of sine and cosine, and end up with the same result.

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 Vaia!

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

By using Laplace transforms, solve the following differential equations subject to the given initial conditions.

y"-4y'+4y=6e2t,y0=y'=0

The speed of a particle on the x axis, x0, is always numerically equal to the square root of its displacement x. If x=0when t=0, find x as a function of t. Show that the given conditions are satisfied if the particle remains at the origin for any arbitrary length of time t0and then moves away; find x for t>t0for this case.

(a)Consider a light beam travelling downward into the ocean. As the beam progresses, it is partially absorbed and its intensity decreases. The rate at which the intensity is decreasing with depth at any point is proportional to the intensity at that depth. The proportionality constant μis called the linear absorption coefficient. Show that if the intensity at the surface is I0, the intensity at a distance s below the surface is I=I0eμs. The linear absorption coefficient for water is of the order of 10.2ft.1(the exact value depending on the wavelength of the light and the impurities in the water). For this value of μ, find the intensity as a fraction of the surface intensity at a depth of 1 ft, ft,ft,mile. When the intensity of a light beam has been reduced to half its surface intensity (I=12I0), the distance the light has penetrated into the absorbing substance is called the half-value thickness of the substance. Find the half-value thickness in terms of μ. Find the half-value thickness for water for the value of μgiven above.

(b) Note that the differential equation and its solution in this problem are mathematically the same as those in Example 1, although the physical problem and the terminology are different. In discussing radioactive decay, we call λthe decay constant, and we define the half-life T of a radioactive substance as the time when N=12N0(compare half-value thickness). Find the relation between λand T.

Prove the general formula L29.

Solve by use of Fourier series. Assume in each case that the right-hand side is a periodic function whose values are stated for one period.

y"+2y'+2y=|x|,-π<x<π.

See all solutions

Recommended explanations on Physics Textbooks

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.

Sign-up for free