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Chapter 4: Q4.3-16E (page 172)

Find a general solution y''+10y'+41y=0

Short Answer

Expert verified

The general solution of the given equationy''+10y'+41y=0isy(t)=e-5t(c1cos(4t)+c2sin(4t)).

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tcos尾t+c2e伪tsin尾t.

02

Finding the roots of the auxiliary equation.

Given differential equation isy''+10y'+41y=0.

Then the auxiliary equation is r2+10r+41=0.

Solve the auxiliary equation to obtain the roots.

role="math" localid="1654071761131" r=-10卤102-4脳1脳412脳1r=-10卤100-1642r=-10卤-642r=-10卤8i2r=-5卤4i

03

Final answer.

Therefore, the general solution is:

y(t)=e-5脳t(c1cos(4t)+c2sin(4t))=e-5t(c1cos(4t)+c2sin(4t))

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Most popular questions from this chapter

Given that y1(t)=14sin2tis a solution toy''+2y'+4y=cos2tandy2(t)=t4-18is a solution torole="math" localid="1654930126913" y''+2y'+4y=t, use the superposition principle to find solutions to the following differential equations:

(a)鈥夆赌夆赌夆赌�y''+2y'+4y=t+cos2t

(b)鈥夆赌夆赌夆赌�y''+2y'+4y=2t-3cos2t

(c)鈥夆赌夆赌夆赌�y''+2y'+4y=11t-12cos2t

Find a general solution.y'''-y''+2y=0

Find a general solution to the differential equation.

y''-y=-11t+1

Find a general solution y''+4y'+8y=0

The auxiliary equation for the given differential equation has complex roots. Find a general solution.4y''+4y'+6y=0
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