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Chapter 4: Q5E (page 199)

In Problems 5 through 8, determine whether Theorem 5 applies. If it does, then discuss what conclusions can be drawn. If it does not, explain why.

t2z''+tz'+z=cost;z(0)=1,z'(0)=0

Short Answer

Expert verified

The differential equation has no unique solution in -鈭�<t<鈭�.

Step by step solution

01

Find the value of p(t),q(t),g(t)

The given differential equation is t2z''+tz'+z=cost.

It can be written as z''+1tz'+1t2z=costt2.

So,p(t)=1t,q(t)=1t2,g(t)=costt2

02

Check the result

From theorem (5) If p(t), q(t), and g(t) are continuous on an interval (a, b) that contains the point t, then for any choice of the initial values Yo鈥�and鈥�鈥�Y1, there exists a unique solution y(1) on the same interval (a, b) to the initial value problems.

Here p(t),q(t), and g(t) are continuous functions in the interval 0<t<鈭�, and the pointt0=0isn't in the interval-鈭�<t<鈭�

Therefore, the differential equation has no unique solution in -鈭�<t<鈭�.

This is the required result.

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Most popular questions from this chapter

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.

Find a general solution.2y"+13y'-7y=0

Find the solution to the initial value problem.

y''+9y=27;鈥夆赌夆赌夆�夆��y(0)=4,鈥夆赌夆赌夆��y'(0)=6

The auxiliary equations for the following differential equations have repeated complex roots. Adapt the "repeated root" procedure of Section 4.2 to find their general solutions:

(a)y''''+2y''+y=0

(b)y''''+4y'''+12y''+16y'+16y=0

Solve the given initial value problem y''+2y'+17y=0;y(0)=1,y'(0)=-1.

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