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

The auxiliary equation for the given differential equation has complex roots. Find a general solution.y''-10y'+26y=0

Short Answer

Expert verified

The auxiliary equation for the given differential equationy''-10y'+26y=0 has complex roots and its general solution is y(t)=e5t(c1cos(t)+c2sin(t)).

Step by step solution

01

Complex conjugate roots.

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

y(t)=c1e伪迟肠辞蝉尾迟+c2e伪迟蝉颈苍尾迟

02

The auxiliary equation.

Assume that y=ertis a solution to the given equation.

Since the given equation is of order two, differentiate role="math" localid="1654061985491" ywith respect to role="math" localid="1654061981089" xtwice:

y'(t)=rerty"(t)=r2ert

Substitutey",y' andyin the given equation to obtain:ert(r2-10r+26)=0.

Then the auxiliary equation isr2-10r+26=0.

03

Finding the roots.

Solve for the roots of the auxiliary equation

r1/2=10卤102-4脳1脳262脳1=10卤100-104=10卤2i=5卤i

04

Final answer.

Since it has a complex conjugate of the form r=伪卤颈尾for 伪=5and 尾=1

Thus, the general solution isy(t)=e5t(c1cos(t)+c2sin(t)).

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