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

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Solutions of Test 1

Jordan Bell

June 2, 2013

1. To have (yµ)0=µy0−2

tµy, we need µ0y=−2

tµy and so

µ0

µ=−2

t.

Integrating,

ln µ=−2 ln t= ln(t−2).

So µ=t−2. Since the left hand side of the equation is equal to (yµ)0, we have

(yt−2)0=tet.

Then we integrate (integrate tetby parts) and get

yt−2=tet−et+C.

As y(1) = 1, we have

1 = e1−e1+C,

so C= 1. Thus the solution is

y=t3et−t2et+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=My−Nt

N=−t5y4−1

t6y4+t=−1

t.

Integrating,

ln µ=−ln t= ln(t−1).

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