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Department of Mathemati s, Mahidol University Kit Tyabandha, PhD Di(cid:11)erential equation st 31 January 2005 De(cid:12)nition1. Adi(cid:11)erentialequation(DE)isanequationwhi hinvolvesderivatives. Anordinary di(cid:11)erential equation (ODE) is a di(cid:11)erential equation in whi h there is exa tly one independent variable. A partial di(cid:11)erential equation (PDE) is one where there are at least two independent variables. ThederivativesofanODEareordinary-,whereasthoseofaPDEarepartialderivatives. x De(cid:12)nition 2. Consider a di(cid:11)erential equation. The order of it is the order of the highest derivative appearing in it. Its degree is the degree of the highest ordered derivative therein. A primitive is a relation between the variables that involves n essential arbitrary onstants, whi h gives rise to a di(cid:11)erential equation of order n. The n onstants are alled essential if they annot be repla ed by a smaller number of onstants. x 000 00 2 0 Example 1. The di(cid:11)erential equation y +3(y ) +2y = sinx is an ordinary di(cid:11)erential 00 2 0 3 equation of order 3 and degree one. The di(cid:11)erential equation (y ) +(y ) +y = 2x is an ODE whi h has an order 2 and degree 2. Problem1. Theproblemof(cid:12)ndingsolutionsofdi(cid:11)erentialequationsisessentiallythatof(cid:12)nding the primitive whi h gave rise to the equation. x 000 2 000 00 Example 2. The di(cid:11)erential equation y = 0 has a primitive y = Ax +Bx+C, y (cid:0)6y + 0 3x 2x x 2 00 2 2 2 2 2 2 11y (cid:0)6y =0 has y=C1e +C2e +C3e , y (y ) +y =r has (x(cid:0)C) +y =r . x De(cid:12)nition 3. Existen e theorems give onditions by whi h one ould determine whether a dif- ferential equation is solvable. A parti ular solution of a di(cid:11)erential equation is one obtained from the primitivebyassigningde(cid:12)nite valuesto theparameters,that isto say,the arbitrary onstants. Asingular solution isasolutionwhi h annotbe obtainedfrom theprimitiveby anymanipulation of the arbitrary onstants. The primitive of a di(cid:11)erential equation is usually alled the general solution of the equation. x De(cid:12)nition 4. A di(cid:11)erential equation is said to be variable separable if an integrating fa tor an be readily found. Su h equation has the form f2(x)(cid:1)g2(y)dx+f2(x)(cid:1)g1(y)dy =0 Through the use of the integrating fa tor 1 f2(x)(cid:1)g2(y) the primitive of this is then Z Z f1(x) g1(y) dx+ dy=C f2(x) g2(y) x De(cid:12)nition 5. A di(cid:11)erential equation of the (cid:12)rst order and (cid:12)rst degree may be written in the form M(x;y)dx+N(x;y)dy=0 If this su h equation admits a solution f(x;y;C)=0 where C is an arbitrary onstant, then there exist in(cid:12)nitely many integrating fa tors xi(x;y) su h that (cid:24)(x;y)[M(x;y)dx+N(x;y)dy℄ =0 is exa t,andthereexisttransformationsofthevariableswhi hrenderthelatterseparated. Butsin e no general rules exist for doing this, the use in pra ti e is still somewhat limited. x st Business Mathemati s, Di(cid:11)erential equation {1{ From 5 Nov 05, as of 31 January, 2006 Department of Mathemati s, Mahidol University Kit Tyabandha, PhD De(cid:12)nition 6. A fun tion f(x;y) is said to be homogeneous of degree n if n f((cid:21)x;(cid:21)y)=(cid:21) f(x;y) x Note 1. The equation (a1x+b1y+ 1)dx+(a2x+b2y+ 2)dy =0 where a1b2(cid:0)a2b1 =0, is redu ed through the transformation dt(cid:0)a1dx a1x+b1y =t and dy = b1 to the form P(x;t)dx+Q(x;t)dt=0 x Note 2. The equation (a1x+b1y+ 1)dx+(a2x+b2y+ 2)dy =0 where a1b2(cid:0)a2b1 6=0, is redu ed through the transformation 0 0 x=x +h and y=y +k inwhi hx=handy=k arethesolutionsoftheequationsa1x+b1y+ 1 =0anda2x+b2y+ 2 =0 into the homogeneous form 0 0 0 0 0 0 (a1x +b1y )dx +(a2x +b2y )dy =0 x Note 3. The equation of the form y(cid:1)f(xy)dx+x(cid:1)g(xy)dy=0 through the transformation z xdz(cid:0)zdx xy =z; y = x; dy = x2 into the form P(x;y)dx+Q(x;z)dz =0 whi h is variable separable. x Bibliography FrankAyres,Jr. TheoryandproblemsofDi(cid:11)erentialEquations. S haum'sOutlineSeries,1981(1952) st Business Mathemati s, Di(cid:11)erential equation {2{ From 5 Nov 05, as of 31 January, 2006