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Class Note for MATH 250A with Professor Lega at UA

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This 2 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Arizona taught by a professor in Fall. Since its upload, it has received 12 views.

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Date Created: 02/06/15
Calculus and Differential Equations l MATH 250 A Differential equations of the form y gXy Human aluauonsoflhe form yquot g M Calculus and Differential Equationsl Separable equations a A first order differential equation is separable if it is of the form 3 mm 0 To solve such an equation we write 1 fy and integrate both sides dy gX dX 0 Of course when we divide by fy we may lose equilibrium solutions which are then singular solutions 9 Example Solve X2y yy 71 Then check your answer Human aluauonsoflhe form y gh yl Calculus and Differential Equationsl Equations with homogeneous coefficients a A differential equation of the form V g X X is said to have homogeneous coefficients 0 To solve such an equation set 2 Z and find an equation X forz d d 717 2 gz7 dxixdxz o The above equation is separable and can then be solved by separation of variables As usual one has to be careful to keep track of singular solutions 2 2 o Example Solve y Then check your answer Xy Differential aluauonsoflhe form yquot g M Calculus and Differential Equationsl Linear equations 0 We now consider Linear first order differential equations equations of the form y pXy qX 0 To solve this equation realize that the left hand side can be written as y PXy egtltP 7 PX dX dix yexp px dxgtgt a As a consequence we have m x X Kmqm dw Kx exp Xpm dt Human aluauonsoflhe form yquot g M Calculus and Differential Equationsl Linear equations continued Bernoulli equations a The function KX is called an integrating factor 0 A first order differential equation of the form 0 Existence and uniqueness theorem for first order linear y pXy qXy equations where n is a positive or negative integer with n 1 is called If p and q are continuous functions of X on the interval ab a Bernoulli equation and if X0 6 ab then there is a unique solution of a To solve this equation make the change of variable y pX y qX u y for X E a b I I I I to obtain a linear equation for M Then solve the equation for 0 Example Solve the equation for an RL electrical circuit u and express the answer in terms of y Ld R E 0 As usual when manipulating differential equations one should E T T tli be careful not to lose or introduce solutions where is the current intensity Et is the voltage R is a 0 Example Solve y ayey3 which is the equation describing resistance and L an inductance a Pitchfork bifurcation as 3 goes through 0 ntial equations of the form t m yi Calculus and D rential Equationsl Differential equations of e form y gm yi Calculus and al Equationsl Uniqueness dy E XVy i a 7 gX7y 0 Existence If g is a continuous function of X and y on the rectangle o Uniqueness If g is a continuous function of X and if there exists a constant k gt 0 such that for all X y1 and y2 in R we RX7 7 lX XOlSav lYYOle have X7 7 X7 S k e 7 where a gt 0 and b gt 0 then there exists a continuously lg yl gl yzll lyl yzl differentiable solution y of the above differential equation on the there eXlStS 3 unique SOlUtlon y to the abOVe differential ixixoi g a for which yX0 yOI Where equation on lXQl g a such that yXo yo The rectangle R and the number a are defined as in the previous theorem a min 7 M Xqiagklgvm This is the Ca uchy Peano theorem This is the Picard Lindelo39f Theorem Difi ntial equations of the form i ri vi Calculus and Differential Equations Differential equations of the ft yquot gi vi Calculus and Dif ntial Equations

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