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Chapter 4: Q45E (page 201)

Find a particular solution to the non-homogeneous equation ty''-(t+1)y'+y=t2e2t, given thatf(t)=etis a solution to the corresponding homogeneous equation.

Short Answer

Expert verified

The solution for the given homogeneous equationty''-(t+1)y'+y=t2e2t is:

y=c1et-c2t+1e2t.

Step by step solution

01

Finding y

Given differential equation isty''-(t+1)y'+y=t2e2t then the standard form of the solution is role="math" localid="1664262459117" y''-t+1ty'+1ty=te2tandf(t)=et is one of the solutions and p(t)=-t+1t.

Theny2=y1(t)鈭�e-鈭�p(t)dty12(t)dt

02

Simplification

Simplify the above

=et鈭�e鈭�1+ttdtet2dt=et鈭�elnt1+te2tdt=et脳鈭�t脳ete2tdt=et脳(-t-1)e-te2t=-t+1e2t

So, the solution is y=c1et-c2t+1e2t.

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Most popular questions from this chapter

A nonhomogeneous equation and a particular solution are given. Find a general solution for the equation. y''+y'=1,鈥夆赌夆赌夆赌夆赌夆赌�yp(t)=t

Swinging Door. The motion of a swinging door with an adjustment screw that controls the amount of friction on the hinges is governed by the initial value problem

滨胃''+产胃'+办胃=0;鈥夆赌夆赌�胃(0)=胃0,鈥夆赌夆赌�胃'(0)=v0,

where胃 is the angle that the door is open,I is the moment of inertia of the door about its hinges,b>0 is a damping constant that varies with the amount of friction on the door,k>0 is the spring constant associated with the swinging door,胃0 is the initial angle that the door is opened, andv0 is the initial angular velocity imparted to the door (see figure). If Iandk are fixed, determine for which values ofb the door will not continually swing back and forth when closing.

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(a)鈥夆赌夆赌夆��y''-y'+y=5sint

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(c)鈥夆赌夆赌夆��y''-y'+y=4sint+18e2t

Find a particular solution to the differential equation.

2z''+z=9e2t
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