Intro - What is a differential equation? A differential...

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Unformatted text preview: What is a differential equation? A differential equation is any equation containing one or more derivatives. Notations of derivatives: ♠ Ordinary derivatives: let y be a function of x, the derivatives of y are denoted by dy ; dx d2 y (2) second derivative: y or 2 ; dx dn y (3) n-th derivative: y (n) or n . dx (1) first derivative: y or ♠ Partial derivatives: let u be a function of x and y , the derivatives of u are denoted by ∂u ∂u , uy or ; ∂x ∂y ∂ 2u ∂ 2u ∂ 2u , uyy or , uxy or . (2) second derivative: uxx or ∂x2 ∂y 2 ∂x∂y (1) first derivative: ux or Examples of differential equations: (1) y = x; (2) y + xy + y = 0; (3) uxx + uyy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í W  FRV W \ í VLQ W \  \ í \  \  \   \   X[[  \  í \ \  \ ILUVW RUGHU 2'( W FRV W HW FRV W  ILUVW RUGHU 2'( VHFRQG RUGHU 2'( IRXUWK RUGHU 2'( XWW  XW VHFRQG RUGHU 3'(  H W WKLUG RUGHU 2'( /LQHDU YV QRQOLQHDU GLIIHUHQWLDO HTXDWLRQV $Q QWK RUGHU RUGLQDU\ GLIIHUHQWLDO HTXDWLRQ LV FDOOHG OLQHDU LI LW FDQ EH ZULWWHQ LQ WKH IRUP \ Q DQí W \ Qí  DQí W \ Qí  «  D W \  D W \  J W  :KHUH WKH IXQFWLRQV D¶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í \  \ HW FRV \ ([HUFLVHV  ±  'HWHUPLQH WKH RUGHU RI HDFK HTXDWLRQ EHORZ $OVR GHWHUPLQH ZKHWKHU HDFK LV D OLQHDU RU QRQOLQHDU HTXDWLRQ  \  W \ FRV W  \  \ í \  HíW \  \ W\  \ W\  \  VHF W  \  \  \  \  í \   \ FRV \ WVLQ W \   HW \   \ í FRW HW \  )RU ZKDW YDOXH V RI Q ZLOO WKH IROORZLQJ HTXDWLRQ EH OLQHDU" \ í \ Q WQ VLQ QW \ OQ W \  \ í H W \ W  \ $QVZHUV  VW RUGHU OLQHDU  UG RUGHU OLQHDU  QG RUGHU QRQOLQHDU  VW RUGHU OLQHDU  WK RUGHU QRQOLQHDU  QG RUGHU OLQHDU  VW RUGHU QRQOLQHDU  WK RUGHU QRQOLQHDU  WK RUGHU OLQHDU  :KHQ Q  RU  WKH HTXDWLRQ LV OLQHDU Some Formulae From Calculus ♠ Derivatives: (tn ) = ntn−1 ; (cos t) = − sin t; (sin t) = cos t; ( e t ) = et ; 1 (ln t) = (t > 0). t ♠ Antiderivatives/Indefinite integrals: tn dt = tn+1 + C, (n = −1); n+1 cos t dt = sin t + C ; sin t dt = − cos t + C ; et dt = et + C ; 1 dt = ln |t| + C (t = 0). t ♠ Some properties of antiderivatives: [f (t) + g (t)] dt = af (t) dt = a Substitution : f (t) dt + g (t) dt; f (t) dt, where a is a constant; t=h(s) f (t) dt = = = == f (h(s))h (s) ds First Order Linear ODE A first order linear ordinary differential equation can be written in the form y’=a(t)y+g(t), where a and g are arbitrary given functions of t. We can also write it as y’+p(t)y=g(t), where p(t) = −a(t). This is called the standard or canonical form of the first order linear equation. We’ll start by attempting to solve a couple of very simple equations of such type: Case 1: a(t) = 0. In this event we get y = g ( t) , which is just a usual integration problem. Then the general solution of this equation is given by y= Comment: The antiderivative g (t) dt, g (t) dt actually represents a collec- tion of functions whose derivative is g (t). Each such function differs from others by one (or more) constant. For example, t2 dt = t3 +C 3 where C is an arbitrary constant. Due to this fact, there are infinitely many solutions for this equation. Examples: Solve the following 1st order linear ODE: (1) y = 10t4 + 3t2 ; 1 (2) y = e3t + , where t = 0; t (3) y = sin t + cos(2t). 3 Answers: (1) y = 2t5 + t3 + C ; 1 (2) y = e3t + ln |t| + C ; 3 1 (3) y = − cos t + sin(2t) + C . 2 Case 2: g (t) = 0. We have y = a(t)y. One trivial solution is given by y = 0. Assume y = 0. Rewrite the equation by dy = a(t)y. dt Move y to the left, and t to the right, we obtain dy = a(t)dt, y then integrate both sides, ln |y | = dy = y a(t) dt, and therefore we get |y | = e a(t) dt . () Comment: The above derivation is a simple example of Separation of Variables, whose basic idea is to separate dependent and independent variables to different sides and then integrate. This method works for many types of equations and shall be used frequently in our course. We will see this technique in later sections. Now let us combine all these solutions in one formula. Pick an antiderivative of a(t), say A(t), then a(t) dt = A(t) + C , and |y | = eC eA(t) = C1 eA(t) , where C1 = eC is an arbitrary but positive number. In fact, we can drop the absolute value sign on the left, then C1 could be any positive 4 or negative (but nonzero) constant. So we rewrite ( ) as y = C1 e A ( t ) , C 1 = 0. When C1 = 0, the above formula indeed gives the trivial solution y = 0. To sum up, the general solution for y = a(t)y is given by y = CeA(t) , for any constant C, where A(t) is an antiderivative of a(t). Examples: Solve the following 1st order linear ODE: (1) y = ty ; 1 (2) y = y, t = 0; t (3) y = 2y − 4. Answers: 12 (1) y = Ce 2 t ; (2) y = C |t|, t = 0; (3) y = Ce2t + 2. (Do the substitution z = y − 2) Examples: Use the method of separation of variables to solve the following ODE: (1) y = 2y − 4; (2) y = t2 e−y . Answers: (1) y = Ce2t + 2; t3 +C . 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