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Unformatted text preview: Most differential equations have an infinite number of solutions. In general, all solutions of a firstorder DE form a family of solutions expressed with a single parameter C . Such a family is called the general solution . A member of the family that results from a specific value of C is called a particular solution. M.I. Bueno Differential Equations and Linear Algebra Example Find the general solution of dy dt = 3 y . Notice that y 0 is a solution. If y 6 = 0 then dy y = 3 dt , Z dy y = Z 3 dt , log  y  = 3 t + c ,  y  = e 3 t + c ,  y  = e c e 3 t ,  y  = De 3 t , D > y = De 3 t . y = Ce 3 t , C R . M.I. Bueno Differential Equations and Linear Algebra Example Find two particular solutions of dy dt = 3 y . Set of solutions: y = Ce 3 t , C R . For C = 0, we get the particular solution y 0. For C = 1, we get the particular solution y = e 3 t . M.I. Bueno Differential Equations and Linear Algebra Newtons cooling law: Find the general solution of dT dt = k ( A T ) , k > , A T < . dT A T = kdt , Z dT A T = Z kdt , We know that R f ( y ) f ( y ) dy = log  f ( y )  + C . Z dT A T = Z kdt , log  A T  = kt + C , log ( T A ) = kt + C , log ( T A ) = kt C , T A = e kt C , T = A + e C e kt , T = A + De kt , D > . M.I. Bueno Differential Equations and Linear Algebra Initialvalue problems We look for a solution to a differential equation that has a specified yvalue y at a given time t . That specified point is called initial value . The combination of a firstorder DE and an initial condition is called Initialvalue problem . dy dt = f ( t , y ) , y ( t ) = y . While a DE generally has a family of solutions, an IVP usually has only one. M.I. Bueno Differential Equations and Linear Algebra Exercise Verify that y = e t / 2 e 3 t satisfies the following initialvalue problem: y + 3 y = e t , y ( ) = 1 / 2 . Recall that d ( e u ) dt = du dt e u . y = e t / 2 ( 3 ) e 3 t = e t / 2 + 3 e 3 t . y + 3 y = ( e t / 2 + 3 e 3 t ) + ( 3 e t / 2 3 e 3 t ) . = ( 1 / 2 + 3 / 2 ) e t = e t . M.I. Bueno Differential Equations and Linear Algebra Exercise Consider the secondorder linear differential equation y 00 y 2 y = 0. Verify that for any constants A and B , y = Ae 2 t + Be t is a solution, taking into account that p = e 2 t and q = e t are both solutions of the differential equation. y = Ap + Bq ....
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This note was uploaded on 03/14/2010 for the course MATH 3C taught by Professor Jacobs during the Fall '08 term at UCSB.
 Fall '08
 JACOBS
 Differential Equations, Equations

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