Characteristic Equations with Complex Roots

Author: John J Weber III, PhD Corresponding Textbook Sections:

Expected Educational Results

Complex Solutions to Characteristic Equations

Let z(t)=u(t)+iv(t) be a solution to ay+by+cy=0, where a,b,cR. Then the real part u(t) and the imaginary part v(t) are real-valued solutions.

If r=α±iβ are complex conjugate solutions to ay+by+cy=0, then two linearly independent solutions are eαtcos(βt) and eαtsin(βt), and the general solution is\newline y(t)=c1eαtcos(βt)+c2eαtsin(βt), where c1 and c2 are arbitrary constants.

Investigation 09

Let y=ert, where r=α±iβ, be solutions to ay+by+cy=0.

  1. Use Euler's formula to rewrite ya=e(α+iβ)t.

  2. Use Euler's formula to rewrite yb=e(αiβ)t.

  3. Evaluate y1=12ya+12yb. Explain why 12 is a valid coefficient for ya and yb.

  4. Evaluate y2=12iya12iyb. Explain why 12i is a valid coefficient for ya and yb.

  5. Explain why y1 and y2 are solutions to ay+by+cy=0.

  6. Explain why y=c1y1+c2y2 is a solution to ay+by+cy=0.

Investigation 10

Suppose the following are roots to the characteristic equation. Find the homogeneous solution, yh(x).

  1. r=2i,2i

  2. r=0,3i,3i

  3. r=1,i3,i3

  4. r=3i,3i,3i,3i

  5. r=1,1,2i,2i

  6. r=12i,1+2i

  7. r=2,2,24i,2+4i

  8. r=32i2,3+2i2

Investigation 11

Solve the following nth-order homogeneous DEs using linear combination of n linearly-independent solutions:

  1. y2y2y=0

  2. y+3y+6y=0

  3. 36y36y+13y=0

  4. y+y+y=0

  5. y+3y=0

  6. y(4)+y2y=0

  7. y(4)+32y+256y=0

  8. y(5)5y+4y=0

  9. y(5)(t)+y(4)(t)+4y(t)+3y(t)+3y(t)=0

Investigation 12

Solve the following nth-order IVPs using linear combination of n linearly-independent solutions:

  1. y+4y+5y=0; y(0)=0 and y(0)=2

  2. y2y+2y=0; y(0)=1 and y(0)=0

  3. y(t)+3y(t)+6y(t)+8y(t)=0, y(0)=1, y(0)=3, y(0)=2

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Last Modified: Monday, 6 September 2020 13:33 EDT