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Chapter 6: Q9E (page 332)

Find a general solution for the differential equation with x as the independent variable:

u'''鈭�9u''+27u'鈭�27u=0

Short Answer

Expert verified

The general solution for the differential equation with x as the independent variable is u(x)=c1e3x+c2xe3x+c3x2e3x

Step by step solution

01

Auxiliary equation:

In the equation,r3鈭�9r3+27r鈭�27=0 , we recognize a complete cube, namely,(r鈭�3)3=0 . Thus, it has just one root x = 3 of multiplicity three.

02

General solution:

The general solution to the given differential equation is given by

u(x)=c1e3x+c2xe3x+c3x2e3x

Hence the final solution is u(x)=c1e3x+c2xe3x+c3x2e3x

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Most popular questions from this chapter

(a) Derive the form y(x)=A1ex+A2e鈭�x+A3cosx+A4sinx for the general solution to the equation y(4)=y, from the observation that the fourth roots of unity are 1, -1, i, and -i.

(b) Derive the form

y(x)=A1ex+A2e鈭�x/2cos(3x2)+A3e鈭�x/2sin(3x2)

for the general solution to the equation y(3)=yfrom the observation that the cube roots of unity are 1,ei2蟺3 , and e鈭�i2蟺3.

Determine whether the given functions are linearly dependent or linearly independent on the interval(0,鈭�) .

(a){e2x,x2e2x,e-x}

(b){exsin2x,xexsin2x,ex,xex}

(c) {2e2x-ex,e2x+1,e2x-3,ex+1}

Use the result of Problem 34 to construct a third-order differential equation for which{x,鈥�sinx,鈥�cosx} is a fundamental solution set.

Find a general solution for the differential equation with x as the independent variable:

y'''鈭�y''+2y=0

Find a general solution for the differential equation with x as the independent variable:

y'''+5y''+3y'鈭�9y=0
See all solutions

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