Lecture 5 Notes

Lecture 5 Notes - Today Solving a second order linear...

Info iconThis preview shows pages 1–16. Sign up to view the full content.

View Full Document Right Arrow Icon
Today • Solving a second order linear homogeneous equation with constant coefFcients • complex roots to the characteristic equation, • repeated roots to the characteristic equation (Reduction of Order). • Connections to matrix algebra. • Solving a second order linear non homogeneous equation.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Reminder: Euler’s formula e i θ = cos θ + i sin θ
Background image of page 2
Complex roots (Section 3.3) • For the general case, , by assuming we get the characteristic equation : ay °° + by ° + cy =0 y ( t )= e rt ar 2 + br + c
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Complex roots (Section 3.3) • For the general case, , by assuming we get the characteristic equation : • When b 2 - 4ac < 0, we get complex roots: ay °° + by ° + cy =0 y ( t )= e rt ar 2 + br + c r 1 , 2 = b ± b 2 4 ac 2 a
Background image of page 4
Complex roots (Section 3.3) • For the general case, , by assuming we get the characteristic equation : • When b 2 - 4ac < 0, we get complex roots: ay °° + by ° + cy =0 y ( t )= e rt ar 2 + br + c r 1 , 2 = b ± b 2 4 ac 2 a = b ± 1 4 ac b 2 2 a
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Complex roots (Section 3.3) • For the general case, , by assuming we get the characteristic equation : • When b 2 - 4ac < 0, we get complex roots: ay °° + by ° + cy =0 y ( t )= e rt ar 2 + br + c r 1 , 2 = b ± b 2 4 ac 2 a = b ± 1 4 ac b 2 2 a = b ± i 4 ac b 2 2 a
Background image of page 6
Complex roots (Section 3.3) • For the general case, , by assuming we get the characteristic equation : • When b 2 - 4ac < 0, we get complex roots: ay °° + by ° + cy =0 y ( t )= e rt ar 2 + br + c r 1 , 2 = b ± b 2 4 ac 2 a = b ± 1 4 ac b 2 2 a = b ± i 4 ac b 2 2 a = b 2 a ± 4 ac b 2 2 a i
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Complex roots (Section 3.3) • For the general case, , by assuming we get the characteristic equation : • When b 2 - 4ac < 0, we get complex roots: ay °° + by ° + cy =0 y ( t )= e rt ar 2 + br + c r 1 , 2 = b ± b 2 4 ac 2 a = b ± 1 4 ac b 2 2 a = b ± i 4 ac b 2 2 a = α ± β i = b 2 a ± 4 ac b 2 2 a i
Background image of page 8
Complex roots (Section 3.3) • Complex roots to the characteristic equation mean complex valued solution to the ODE:
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Complex roots (Section 3.3) • Complex roots to the characteristic equation mean complex valued solution to the ODE: y 1 ( t ) = e ( α + β i ) t
Background image of page 10
Complex roots (Section 3.3) • Complex roots to the characteristic equation mean complex valued solution to the ODE: = e α t e i β t y 1 ( t ) = e ( α + β i ) t
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Complex roots (Section 3.3) • Complex roots to the characteristic equation mean complex valued solution to the ODE: = e α t e i β t = e α t (cos( β t )+ i sin( β t )) y 1 ( t ) = e ( α + β i ) t
Background image of page 12
Complex roots (Section 3.3) • Complex roots to the characteristic equation mean complex valued solution to the ODE: = e α t e i β t = e α t (cos( β t )+ i sin( β t )) y 1 ( t ) = e ( α + β i ) t y 2 ( t ) = e ( α β i ) t
Background image of page 13

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Complex roots (Section 3.3) • Complex roots to the characteristic equation mean complex valued solution to the ODE: = e α t e i β t = e α t (cos( β t )+ i sin( β t )) = e α t e i β t y 1 ( t ) = e ( α + β i ) t y 2 ( t ) = e ( α β i ) t
Background image of page 14
Complex roots (Section 3.3) • Complex roots to the characteristic equation mean complex valued solution to the ODE: = e α t e i β t = e α t (cos( β t )+ i sin(
Background image of page 15

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 16
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 108

Lecture 5 Notes - Today Solving a second order linear...

This preview shows document pages 1 - 16. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online