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Chapter 1: Q19E (page 1)

In Problem 19, solve the given initial value problem

y'''鈭�y''鈭�4y'+4y=0y(0)=鈭�4y'(0)=鈭�1y''(0)=鈭�19

Short Answer

Expert verified

The solution is Cy(t)=et鈭�3e2t鈭�2e鈭�2t

Step by step solution

01

Basic differentiation

The Sum rule says the derivative of a sum of functions is the sum of their derivatives. The Difference rule says the derivative of a difference of functions is the difference of their derivatives.

02

Solving by basic differentiation:

We will do the following question on the basis of basic differentiation ;

r3鈭�r2鈭�4r+4=0r3鈭�4r2+7r鈭�6=(r鈭�1)(r鈭�2)(r+2)=0y(t)=c1et+c2e2t+c3e鈭�2ty'(t)=c1et+2c2e2t鈭�2c3e鈭�2ty''(t)=c1et+4c2e2t+4c3e鈭�2ty(0)=鈭�4y'(0)=鈭�1y''(0)=鈭�19y(t)=et鈭�3e2t鈭�2e鈭�2t

Hence, the final answer is:

y(t)=et鈭�3e2t鈭�2e鈭�2t

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Most popular questions from this chapter

In Problems 14鈥�24, you will need a computer and a programmed version of the vectorized classical fourth-order Runge鈥揔utta algorithm. (At the instructor鈥檚 discretion, other algorithms may be used.)鈥�

Using the vectorized Runge鈥揔utta algorithm for systems withh=0.175, approximate the solution to the initial value problemx'=2x-y;x(0)=0,y'=3x+6y;y(0)=-2 att=1.

Compare this approximation to the actual solution.

Consider the differential equation dpdt=p(p-1)(2-p) for the population p (in thousands) of a certain species at time t.

猞� Sketch the direction field by using either a computer software package or the method of isoclines.

猞� If the initial population is 4000 [that is, p0=4], what can you say about the limiting population limt鈫�鈭�P(t)?

猞� If p(0)=1.7, what is limt鈫�鈭�P(t)?

猞� If p(0)=0.8, what is limt鈫�鈭�P(t)?

猞� Can a population of 900 ever increase to 1100?

Question:(a) Use the general solution given in Example 5 to solve the IVP. 4x"+e-0.1tx=0,x(0)=1,x'(0)=-12.Also use J'0(x)=-J1(x) and Y'0(x)=-Y1(x)=-Y1(x)along withTable 6.4.1 or a CAS to evaluate coefficients.

(b) Use a CAS to graph the solution obtained in part (a) for.

The motion of a set of particles moving along the x鈥慳xis is governed by the differential equation dxdt=t3-x3, where xt denotes the position at time t of the particle.

猞� If a particle is located at x=1 when t=1 , what is its velocity at this time?

猞� Show that the acceleration of a particle is given by d2xdt2=3t2-3t3x2+3x5.

猞� If a particle is located at x=2when t=2.5, can it reach the location x=1at any later time?

[Hint: t3-x3=(t-x)(t2+xt+x2).]

In Problems 10鈥�13, use the vectorized Euler method with h = 0.25 to find an approximation for the solution to the given initial value problem on the specified interval.

t2y''+y=t+2;y(1)=1,y'(1)=-1on[1,2]
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