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Chapter 4: Q2E (page 164)

Question: Find a geneQuestion: Find a general solution to the given differential equation.ral solution to the given differential equation.

y''+6y'+9y=0

Short Answer

Expert verified

Answer

The general solution of the given equation isy=c1e-3t+c2te-3t.

Step by step solution

01

Write the auxiliary equation of the given differential equation.

The differential equation isy''+6y'+9y=0.y''+6y'+9y=0.

The auxiliary equation for the above equationm2+6m+9=0.

02

Find the roots of the auxiliary equation. 

Solve the auxiliary equation,

m2+6m+9=0m2+3m+3m+9=0mm+3+3m+3=0m+3m+3=0

The roots of the auxiliary equation arem1=-3,鈥�&鈥�m2=-3.

If the auxiliary equation has repeated real roots, then the general solution is given as;

y=c1em1t+c2tem2t

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Most popular questions from this chapter

Decide whether the method of undetermined coefficients together with superposition can be applied to find a particular solution of the given equation. Do not solve the equation.

2y''+3y'-4y=2t+sin2t+3

Decide whether or not the method of undetermined coefficients can be applied to find a particular solution to the given equation. y''+2y'-y=t-1et

The auxiliary equation for the given differential equation has complex roots. Find a general solution y''+9y=0.

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

Find a general solution z"+10z'+25z=0
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