MAP
de-t2-a

# Of the differential equation y y 4sec x obviously the

• Notes
• 4

This preview shows pages 2–3. Sign up to view the full content.

, of the differential equation y y 4sec( x ). Obviously the driving function here is NOT a UC function. Thus, we must use variation of parameters to nab the culprit. Corresponding Homogeneous: F.S. = {sin( x ), cos( x )}. y y 0. If y p = v 1 cos( x ) + v 2 sin( x ) then v 1 and v 2 are solutions to the following system: Solving the system yields v 1 = -4tan( x) and v 2 = 4. Thus, by integrating, we obtain v 1 4ln sec( x ) c and v 2 4 x d . Thus, a particular integral of the ODE above is y p v 1 cos( x ) v 2 sin( x ) 4ln sec( x ) cos( x ) 4 x sin( x ). ______________________________________________________________________ 4. (10 pts.) Set up the correct linear combination of undetermined coefficient functions you would use to find a particular integral, y p , of the O.D.E. y 4 y 5 y x 2 e 2 x sin( x ) e 2 x . [ Warning: (a) If you skip a critical initial step, you will get no credit!! (b) Do not waste time attempting to find the numerical values of the coefficients!! ] First, the corresponding homogeneous equation is y 4 y 5 y 0. which has an auxiliary equation given by 0 = m 2 - 4 m + 5. Thus, m = 2 + i or m = 2 - i , and a fundamental set of solutions for the corresponding homogeneous equation is { exp(2 x )cos( x ), exp(2 x )sin( x ) }. Taking this into account, we may now write y p ( x ) Ax 2 e 2 x Bxe 2 x Ce 2 x Dx cos( x ) e 2 x Ex sin( x ) e 2 x or something equivalent. ______________________________________________________________________ Silly 10 Point Bonus: Let f ( x ) = x and g ( x ) = sin( x ). (a) It is trivial to obtain a 4th order homogeneous linear constant coefficient ordinary differential equation with f and g as solutions. Do so. (b) It’s only slightly messier to obtain a 2nd order homogeneous linear ordinary differential equation with { f , g } as a fundamental set of solutions. Do so.

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

This is the end of the preview. Sign up to access the rest of the document.
• Fall '08
• STAFF

{[ 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