Chapter1Section6

Chapter1Section6 - 1.6 Linear First Order Equations The Extended Chain Rule x v x then Recall that if y Form I dy F ax dx by Integrate 1 dy dx dv

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1.6: Linear First Order Equations ± The Extended Chain Rule Recall that if y = Β H x , v H x LL , then dy dx = ¶Β x ± dx dx + ¶Β v ± dv dx = Β x + Β v ± dv dx . ± Form I: dy dx = F H a x + b y + c L . Substitution: v = a x + b y + c . Integrate @ 1 ± H 4 Sqrt @ v D + 3 L , v D 1 8 J 3 + 4 v - 3 Log B 3 + 4 v FN ª Example: Consider the differential equation dy dx = 3 x + 4 y + 7 . A. Solve the differential equation. B. List any restrictions for the solution to be meaningful.
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We show the direction field along with the domain restriction. theDE = Sqrt @ 3 x + 4 y + 7 D ; ivp = 88 0, 1 < , 8 0, - 1 < , 8 - 1.5, 1 < , 8 2, 2 < , 8 3.5, 0 < , 8 - 3, 2 << ; Show @ VectorPlot @8 1, theDE < , 8 x, - 4, 4 < , 8 y, - 3, 3 < , FrameLabel 8 x, y < , Axes True, VectorScale 8 Small, Tiny, None < , VectorStyle Gray, StreamScale Full, StreamStyle 8 Blue, Thick, "Line" < , StreamPoints 8 ivp < , PlotLabel "Direction Field", Epilog 8 PointSize @ Large D , Point @ ivp D< D , Plot @ - 3 ± 4 x - 7 ± 4, 8 x, - 4, 4 < , PlotStyle 8 Red, Dashed, Thick <D D - 4 - 2 0 2 4 - 3 - 2 - 1 0 1 2 3 x y Direction Field 2 Chapter1Section6.nb
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± Definition A h o m o g e n e o u s first order differential equation can be written in the form dy dx = F J y x N . ± Form II: Homogeneous DEs. Substitution: v = y x . Notice that when v = y x , we have y = xv, and hence dy dx = v + x ± dv dx since v is a function of x . Hence dy dx = F J y x N ² v + x dv dx = F H v L ² x dv dx = F H v L - v , Notice that the above is a separable equation! ª Example Consider the initial value problem H x + 2 y L y ' = y , y H 2 L = 1. A. Solve the initial value problem. B. List any restrictions. Chapter1Section6.nb 3
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Look! DSolve threatens to coughs up a hairball! It is warning us that inverse functions were used on something that may not be a function! sol1 = y @ x D ± . DSolve @8H x + 2 y @ x DL y' @ x D ± y @ x D , y @ 2 D ± 1 < , y @ x D , x D InverseFunction::ifun: Inverse functions are being used. Values may be lost for multivalued inverses. ± Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. ± Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. ± : x 2 ProductLog A ª x 2 E > Anyone heard of the ProductLog function? it is the principal solution for w in z = we w . If z ? - 1 ± e , then you have real solutions. 4 Chapter1Section6.nb
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= y ± H x + 2 y L ; ivp = 88 0, 1 ± 2 < , 8 0, - 1 < , 8 2, 2 < , 8 - 2, 2 < , 8 3, - 1 << ; Show @ VectorPlot @8 1, theDE < , 8 x, - 4, 4 < , 8 y, - 3, 3 < , FrameLabel 8 x, y < , Axes True, VectorScale 8 Small, Tiny, None < , VectorStyle Gray, StreamScale Full, StreamStyle 8 Blue, Thick, "Line" < , StreamPoints 8 ivp < , PlotLabel "Direction Field", Epilog 8 PointSize @ Large D , Point @ ivp D< D , Plot @ sol1, 8 x, - 4, 4 < , PlotStyle 8 Red, Dashed, Thick <D D - 4 - 2 0 2 4 - 3 - 2 - 1 0 1
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This note was uploaded on 01/16/2012 for the course MATH 306 taught by Professor Keithemmert during the Fall '11 term at Tarleton.

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Chapter1Section6 - 1.6 Linear First Order Equations The Extended Chain Rule x v x then Recall that if y Form I dy F ax dx by Integrate 1 dy dx dv

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