MAT187H1 Lecture 14: 8.4 First Order Linear DEs & 8.5 Modelling with DEs

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MAT187H1Lecture148.4FirstOrderLinearDEs&8.5ModellingwithDEs
Review
TosolveDEofform (x)y(x)
dx
dy +P=Q
Define(IntegratingFactor)(x)μ = e(x)dx
P
Multiplybothsidesby (x)μ = e(x)dx
P
yy) (x)Q(x)dx μ =
d=
μ
(x)Q(x)dxy = μ−1
μ
NoteWhenevaluating ,don’tneedintegratingconstant(x)μ
Why?
Say (x)e eμ = e(x)dx
P=eR(x)+k=ek R(x)=cR(x)
Whenmultiplyingbothsidesby :(x)μ
e[ (x)y]e[Q(x)] cancelsoutc R(x)
dx
dy +P=cR(x)c
Example tanx)y ecx, ,P(x)anx
dx
dy + ( = s2
π<x<2
π = t
(x)ecxμ = eanxdx
t=eln|secx|=s
Note: secx>0
Multiplyby :(x)μ
ecx(tanx)y)ec xs dx
dy + ( = s2
((secx)y)ec xdxd =s2
 (ysecx)ec xdx
d=
s2
 secx anxy =t+C
 y1
cosx =sinx
cosx +C
inx cosxy =s+C
(check: )(ysecx)ecx (secxtanx)HS
d
dx =sdx
dy +yL
8.4SpecialFirstOrderLinearDE&8.5ModellingwithDE’s
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