Reduction of Order

May 8th, 2015
Studypool Tutor
Price: $10 USD

Tutor description

Consider a linear, second-order, homogeneous equation in standard form, y 00 + p(x)y 0 + q(x)y 0 = 0: Suppose that one solution y1 is known. Then a second, independent solution y2 is obtained by letting y2 (x) = u(x)y1 (x); u(x) to be determined

Word Count: 326
Showing Page: 1/5
Math(diff. equations) notes #1MATH 1005A - Notes 1Reduction of OrderConsider a linear, second-order, homogeneous equation in standard form, y 00 + p(x)y 0 + q(x)y 0 = 0:Suppose that one solution y1 is known. Then a second, independent solution y2 is obtained by letting y2 (x) = u(x)y1 (x); u(x) to be determined.00000y2 = uy1 ) y2 = u0 y1 + uy1 ; y 00 = u00 y1 + 2u0 y1 + uy1 ;so y2 is a solution if and only if0000[u00 y1 + 2u0 y1 + uy1 ] + p(x)[u0 y1 + uy1 ] + q(x)[uy1 ] = 0;i.e.,0000u[y1 + p(x)y1 + q(x)y1 ] + u00 y1 + 2u0 y1 + p(x)u0 y1 = 0:000Since y1 is a solution, y1 + p(x)y1 + q(x)y1 = 0. Thus, y2 is a solution if and only if 0u00 y1 + u0 [2y1 + p(x)y1 ] = 0;i.e.,u002y 0 + p(x)y1y0= 1= 2 1 p(x):u0y1y1Integration with respect to x then gives0ln ju j = 2 ln jy1 j Zp(x) dx:Taking the exponential of both sides and using the fact that e2 ln jy1 j = eln jy1 j obtainju0 j =2=1, we2y11 R p(x) dx1 Re; or u0 = 2 e p(x) dx:2y1y1Taking the plus sign and integrating once more, we obtainZ1 R p(x) dxu(x) =edx:2y1Since1 R p(x) dxe6= 0;2y1u(x) is not a constant, hence y1 = erx and y2 = u(x)erx are linearly independent. For the equation ay 00 + by 0 + cy = 0 with b2 4ac = 0; y = erx ) ar2 + br + c = 0 ) pbbb b2 4ac=) y1 = erx = e 2a x :r=2a2a2In standard form, the equation isbcby 00 + y 0 + y = 0; with p(x) = :aaaThus, by reduction of order,

Review from student

Studypool Student
" Thanks, good work "
Ask your homework questions. Receive quality answers!

Type your question here (or upload an image)

1825 tutors are online

Brown University





1271 Tutors

California Institute of Technology




2131 Tutors

Carnegie Mellon University




982 Tutors

Columbia University





1256 Tutors

Dartmouth University





2113 Tutors

Emory University





2279 Tutors

Harvard University





599 Tutors

Massachusetts Institute of Technology



2319 Tutors

New York University





1645 Tutors

Notre Dam University





1911 Tutors

Oklahoma University





2122 Tutors

Pennsylvania State University





932 Tutors

Princeton University





1211 Tutors

Stanford University





983 Tutors

University of California





1282 Tutors

Oxford University





123 Tutors

Yale University





2325 Tutors