Main Theorem for Nonhomogeneous DE Variation of Parameters 5 Variation of

# Main theorem for nonhomogeneous de variation of

• 7

This preview shows page 6 - 7 out of 7 pages.

Main Theorem for Nonhomogeneous DE Variation of Parameters 5 Variation of Parameters: It follows that u 0 1 ( t ) = det 0 y 2 ( t ) g ( t ) y 0 2 ( t ) W [ y 1 , y 2 ]( t ) and u 0 2 ( t ) = det y 1 ( t ) 0 y 0 1 ( t ) g ( t ) W [ y 1 , y 2 ]( t ) Solving this, we obtain: u 0 1 ( t ) = - y 2 ( t ) g ( t ) W [ y 1 , y 2 ]( t ) and u 0 2 ( t ) = y 1 ( t ) g ( t ) W [ y 1 , y 2 ]( t ) , which can be integrated. Joseph M. Mahaffy, h [email protected] i Lecture Notes – Second Order Linear Equations — (22/27) Cauchy-Euler Equation Review Variation of Parameters Motivating Example Technique of Variation of Parameters Main Theorem for Nonhomogeneous DE Variation of Parameters 5 Variation of Parameters: The equations for u 0 1 and u 0 2 are integrated yielding u 1 ( t ) = - Z y 2 ( t ) g ( t ) W [ y 1 , y 2 ]( t ) dt + c 1 and u 2 ( t ) = Z y 1 ( t ) g ( t ) W [ y 1 , y 2 ]( t ) dt + c 2 If these integrals can be evaluated, then the general solution can be written y ( t ) = u 1 ( t ) y 1 ( t ) + u 2 ( t ) y 2 ( t ) , Otherwise, the solution is given in its integral form Joseph M. Mahaffy, h [email protected] i Lecture Notes – Second Order Linear Equations Part 2 - Nonhomogeneous — (23/27) Cauchy-Euler Equation Review Variation of Parameters Motivating Example Technique of Variation of Parameters Main Theorem for Nonhomogeneous DE Variation of Parameters Theorem Consider the nonhomogeneous equation y 00 + p ( t ) y 0 + q ( t ) y = g ( t ) , Theorem If the functions p , q , and g are continuous on an open interval I , and if y 1 and y 2 form a fundamental set of solutions of the homogeneous equation. Then a particular solution of the nonhomogeneous problem is y p ( t ) = - y 1 ( t ) Z t t 0 y 2 ( s ) g ( s ) W [ y 1 , y 2 ]( s ) ds + y 2 ( t ) Z t t 0 y 1 ( s ) g ( s ) W [ y 1 , y 2 ]( s ) ds, where t 0 I . The general solution is y ( t ) = c 1 y 1 ( t ) + c 2 y 2 ( t ) + y p ( t ) . Joseph M. Mahaffy, h [email protected] i Lecture Notes – Second Order Linear Equations — (24/27)

Subscribe to view the full document.

Cauchy-Euler Equation Review Variation of Parameters Motivating Example Technique of Variation of Parameters Main Theorem for Nonhomogeneous DE Variation of Parameters Example 1 Example: Solve the differential equation t 2 y 00 - 2 y = 3 t 2 - 1 , t > 0 As always, first solve the homogeneous equation: t 2 y 00 - 2 y = 0 , which is a Cauchy-Euler Equation Attempt solution y ( t ) = t r , giving t r [ r ( r - 1) - 2] = t r ( r 2 - r - 2) = t r ( r - 2)( r + 1) = 0 , so r = - 1 and r = 2. This gives the homogeneous solution y c ( t ) = c 1 y 1 ( t ) + c 2 y 2 ( t ) = c 1 t - 1 + c 2 t 2 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Second Order Linear Equations Part 2 - Nonhomogeneous — (25/27) Cauchy-Euler Equation Review Variation of Parameters Motivating Example Technique of Variation of Parameters Main Theorem for Nonhomogeneous DE Variation of Parameters Example 2 Example: The differential equation t 2 y 00 - 2 y = 3 t 2 - 1 , t > 0 has homogeneous solutions y 1 ( t ) = t - 1 and y 2 ( t ) = t 2 Compute the Wronskian W [ t - 1 , t 2 ]( t ) = det t - 1 t 2 - t - 2 2 t = 3 To use the Variation of Parameters , we rewrite the DE y 00 - 2 t 2 y = 3 - 1 t 2 = g ( t ) , t > 0 , Joseph M. Mahaffy, h [email protected] i Lecture Notes – Second Order Linear Equations — (26/27) Cauchy-Euler Equation Review Variation of Parameters Motivating Example Technique of Variation of Parameters Main Theorem for Nonhomogeneous DE Variation of Parameters Example 3 Example: From the theorem above, a particular solution satisfies y p ( t ) = - y 1 ( t ) Z t t 0 y 2 ( s ) g ( s ) W [ y 1 , y 2 ]( s ) ds + y 2 ( t ) Z t t 0 y 1 ( s ) g ( s ) W [ y 1 , y 2
• Fall '08
• staff

### 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