Lecture 4 Notes

# Lecture 4 Notes - Today Independence of functions and the...

This preview shows pages 1–12. Sign up to view the full content.

Today • Independence of functions and the Wronskian • Distinct roots of the characteristic equation • Review of complex numbers • Complex roots of the characteristic equation

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

View Full Document
Homog. eq. with constant coeff. (Section 3.1) • For the general case, , by assuming we get the characteristic equation : ay + by + cy = 0 y ( t ) = e rt ar 2 + br + c = 0
Homog. eq. with constant coeff. (Section 3.1) • For the general case, , by assuming we get the characteristic equation : • There are three cases. ay + by + cy = 0 y ( t ) = e rt ar 2 + br + c = 0

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

View Full Document
Homog. eq. with constant coeff. (Section 3.1) • For the general case, , by assuming we get the characteristic equation : • There are three cases. i. Two distinct real roots: b 2 - 4ac > 0. ( r 1 r 2 ) ay + by + cy = 0 y ( t ) = e rt ar 2 + br + c = 0
Homog. eq. with constant coeff. (Section 3.1) • For the general case, , by assuming we get the characteristic equation : • There are three cases. i. Two distinct real roots: b 2 - 4ac > 0. ( r 1 r 2 ) ii.A repeated real root: b 2 - 4ac = 0. ay + by + cy = 0 y ( t ) = e rt ar 2 + br + c = 0

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

View Full Document
Homog. eq. with constant coeff. (Section 3.1) • For the general case, , by assuming we get the characteristic equation : • There are three cases. i. Two distinct real roots: b 2 - 4ac > 0. ( r 1 r 2 ) ii.A repeated real root: b 2 - 4ac = 0. iii.Two complex roots: b 2 - 4ac < 0. ay + by + cy = 0 y ( t ) = e rt ar 2 + br + c = 0
Homog. eq. with constant coeff. (Section 3.1) • For the general case, , by assuming we get the characteristic equation : • There are three cases. i. Two distinct real roots: b 2 - 4ac > 0. ( r 1 r 2 ) ii.A repeated real root: b 2 - 4ac = 0. iii.Two complex roots: b 2 - 4ac < 0. • For case i, we get and . ay + by + cy = 0 y ( t ) = e rt ar 2 + br + c = 0 y 1 ( t ) = e r 1 t y 2 ( t ) = e r 2 t

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

View Full Document
Homog. eq. with constant coeff. (Section 3.1) • For the general case, , by assuming we get the characteristic equation : • There are three cases. i. Two distinct real roots: b 2 - 4ac > 0. ( r 1 r 2 ) ii.A repeated real root: b 2 - 4ac = 0. iii.Two complex roots: b 2 - 4ac < 0. • For case i, we get and . • Do our two solutions cover all possible ICs? That is, can we use them to form a general solution ? ay + by + cy = 0 y ( t ) = e rt ar 2 + br + c = 0 y 1 ( t ) = e r 1 t y 2 ( t ) = e r 2 t
Independence and the Wronskian (Section 3.2) • Example: Suppose and are two solutions to some equation. Can we solve ANY initial condition . . with these two solutions? • Solve this system for C 1 , C 2 ... • Can’t do it. Why? y 2 ( t ) = e 2 t 3 y 1 ( t ) = e 2 t +3 y (0) = y 0 , y (0) = v 0

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

View Full Document
Independence and the Wronskian (Section 3.2) • Example: Suppose and are two solutions to some equation. Can we solve ANY initial condition . . with these two solutions? • Solve this system for C 1 , C 2 ... • Can’t do it. Why? y 2 ( t ) = e 2 t 3 y 1 ( t ) = e 2 t +3 y (0) = y 0 , y (0) = v 0 y ( t ) = C 1 e 2 t +3 + C 2 e 2 t 3
Independence and the Wronskian (Section 3.2) • Example: Suppose and are two solutions to some equation. Can we solve ANY initial condition . . with these two solutions?

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern