MAT244H1 Study Guide - Midterm Guide: Wronskian, Jordan Bell
27 views3 pages
Mat 244 Solved Examples Relevant to Test 2
July 2, 2013
1. Let y1(x) = sin(ecos sin log x), and let y2(x) = −sin(ecos sin log x). Let
W(y1, y2) be their Wronskian. Calculate W(1).
Solution. We don’t have to compute the derivative of y1and y2. If functions
are linearly dependent, then their Wronskian is 0. y2is −1 times y1, so they
are linearly dependent. Thus W(x) = 0 for all x, in particular, W(1) = 0.
2. Suppose that b2= 4ac, and let
Let Ly =ay00 +by0+cy.
Show that y1(t) = ert and y2(t) = tert are solutions of the diﬀerential
equation Ly = 0.
Solution. First we’ll check that y1is a solution. y0
1=rert and y00
Ly1=ar2ert +brert +cert
=ert ar2+br +c
because ris a root of ar2+br +c. Thus Ly1= 0.
2=ert +rtert , and y00
2=rert +rert +r2tert = 2rert +r2tert .
Ly2=a(2rert +r2tert ) + b(ert +rtert) + ctert
=tert ar2+br +c+ert (2ar +b)
= 0 + ert (2ar +b).
What is crucial is that r=−b
2a, and hence Ly2= 0. Therefore, y1and y2are
both solutions of Ly = 0.
3. (This is problem 16, section 3.6) Let y1=etand y2=t. Show that are
(1 −t)y00 +ty0−y= 0,
and ﬁnd a solution of the diﬀerential equation
(1 −t)y00 +ty0−y= 2(t−1)2e−t.