MAT244H1 Study Guide - Midterm Guide: Jordan Bell, Integrating Factor, Integral Equation

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Published on 25 Oct 2018
School
Department
Course
Professor
Solutions of Test 1
Jordan Bell
June 2, 2013
1. To have (yµ)0=µy02
tµy, we need µ0y=2
tµy and so
µ0
µ=2
t.
Integrating,
ln µ=2 ln t= ln(t2).
So µ=t2. Since the left hand side of the equation is equal to (yµ)0, we have
(yt2)0=tet.
Then we integrate (integrate tetby parts) and get
yt2=tetet+C.
As y(1) = 1, we have
1 = e1e1+C,
so C= 1. Thus the solution is
y=t3ett2et+Ct2.
2. M=t5y5and N=t6y4+t. If we want µM +µN y0= 0 to be exact, then
(µM)y= (µN)t,
i.e.
µyM+µMy=µtN+µNt.
Try an integrating factor that is only a function of t, so µy= 0. Then
µMy=µtN+µNt,
so µt
µ=MyNt
N=t5y41
t6y4+t=1
t.
Integrating,
ln µ=ln t= ln(t1).
1
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