Chapter1Section4 - 1.4 Separable Equations and Applications...

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1.4: Separable Equations and Applications Basic Definitions The first order differential equation dy dx = H H x , y L is called s e p a r a b l e if H H x , y L = g H x L h H y L . Recall: If y = f H x L , then the d i f f e r e n t i a l of y is dy = f ' H x L dx, so we can represent the derivative as f ' H x L = dy dx = Quotient of dy and dx = dy dx = Derivative of y with respect to x . Theorem Suppose dy dx = g H x L h H y L (i.e. we have a separable differential equation). Then dy h I y M = g H x L dx dy h I y M = g H x L x + C . Example Consider the differential equation dy dx = 3 y . A. Solve the initial value problem when y H 0 L = 1. B. Solve the initial value problem when y H 0 L = 1. C. Create a direction field showing your two solutions.
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theDE = 3y; ivp = 88 0, 1 < , 8 0, - 1 << ; VectorPlot @8 1, theDE < , 8 x, - 2, 1 < , 8 y, - 3, 3 < , FrameLabel fi 8 x, y < , Axes fi True, VectorScale fi 8 Small, Tiny, None < , VectorStyle fi Gray, StreamScale fi Full, StreamStyle fi 8 Blue, Thick, "Line" < , StreamPoints fi 8 ivp < , PlotLabel fi "Direction Field", Epilog fi 8 PointSize @ Large D , Point @ ivp D< D - 2.0 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 - 3 - 2 - 1 0 1 2 3 x y Direction Field 2 Chapter1Section4.nb
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Partial Fractions Example Remember Partial Fraction Decomposition? 1 1 - x 2 = 1 I 1 - x M I 1 + x M = A 1 - x + B 1 + x Clear Fractions by multiplying by I 1 - x 2 M 1 = A H 1 + x L + B H 1 - x L 1 = A + B + H A - B L x Solve the system of equations 1 = A + B 0 = A - B A = 1 2 , B = 1 2 .
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