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Chapter 6: Q13E (page 332)

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

y(4)+4y''+4y=0

Short Answer

Expert verified

The general solution for the differential equation with x as the independent variable is

y(x)=c1cos(2x)+c2xcos(2x)+c3sin(2x)+c4xsin(2x)

Step by step solution

01

Auxiliary equation:

The auxiliary equation in this problem is r4+4r2+4=0. This can be factored as (r2+2)2=0. Therefore this equation has roots r=2i,−2i,2i,−2i, which we see are repeated and complex.

02

General solution:

The general solution to the given equation is given by

y(x)=c1cos(2x)+c2xcos(2x)+c3sin(2x)+c4xsin(2x)

Hence the final solution is y(x)=c1cos(2x)+c2xcos(2x)+c3sin(2x)+c4xsin(2x)

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Most popular questions from this chapter

Find a general solution to

y(4)+2y'''−3y''−y'+12y=0

by using Newton’s method (Appendix B) or some othernumerical procedure to approximate the roots of the auxiliaryequation.

Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.

{sinx, cosx, tanx}on(-π2, π2)

use the method of undetermined coefficients to determine the form of a particular solution for the given equation.

y'''+y''-5y'+3y=e-x+sinx

Constructing Differential Equations. Given three functions that f1(x), f2(x),f3(x)are each three times differentiable and whose Wronskian is never zero on (a, b), show that the equation

|f1(x)f2(x)f3(x)yf1'(x)f2'(x)f3'(x)y'f1''(x)f2''(x)f3''(x)y''f1'''(x)f2'''(x)f3''(x)y'''|=0

is a third-order linear differential equation for which{f1, f2 f3} is a fundamental solution set. What is the coefficient of y‴ in this equation?

Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.

{x-1, x12, x}on(0, ∞)
See all solutions

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