# MAT244H1 Study Guide - Midterm Guide: Wronskian, Jordan Bell

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Mat 244 Solved Examples Relevant to Test 2

Jordan Bell

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

r=−b

2a.

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

1=r2ert .

Then,

Ly1=ar2ert +brert +cert

=ert ar2+br +c

=ert ·0

= 0,

because ris a root of ar2+br +c. Thus Ly1= 0.

y0

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

solutions of

(1 −t)y00 +ty0−y= 0,

and ﬁnd a solution of the diﬀerential equation

(1 −t)y00 +ty0−y= 2(t−1)2e−t.

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