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Chapter 4: Q4.3-1E (page 172)

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

Short Answer

Expert verified

The auxiliary equation for the given differential equation y''+9y=0has complex roots and its general solution is y(t)=c1cos3t+c2sin3t.

Step by step solution

01

Complex conjugate roots.

If the auxiliary equation has complex conjugate roots 伪卤i尾, then the general solution is given as:

y(t)=c1e伪tcos尾t+c2e伪tsin尾t.

02

Finding the roots of the auxiliary equation.

Given differential equation isy"+9y=0.

Then the auxiliary equation is r2+9=0.

Finding the roots of the auxiliary equation;

r2+9=0r2=-9r=卤3i

03

Final answer.

Therefore, the general solution is:

y(t)=e0脳t(c1cos(3t)+c2sin(3t))=c1cos3t+c2sin3t

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