MATH 202 Useful File, Fall 1998

Section L.1 e Two types of D.E. dus dv 1. Ordinary D.E. ex! ——-—-=x ; , ea dx 2. Partial D.E. ex: y =-> e The order of the highest order derivative is called the order of the D.. e ADL. is oe to be linear if it can be written in the form ar? » (X)¥ = B(x) © Two properties must be noted 3.The dependent variable y and all its derivatives are of the first degree ex: y”- 2y’+ y = 0 (linear), (yy - 2y’+y = 0 (non linear). 4.Each coefficient depends only on the independent variable x. ex: yy’-2y’=x (non linear) e An O.D.E. that can be written in the form y = f(x) is said explicit. -A relation G(x,y) = 0 is said to be an implicit solution of an O.D.E. provided | it defines one or more explicit solutions on I. ex: For -2<x <2 x’ +y’ - 4 = Ois on implicit solution of the D.E. gy Xx x? +y?—4=0 has two. explicit dx y differentiable functions y =+V4—x? on the interval (-2,2). Section 2.2 Theorem 2.1: Let R be a rectangular region in the x-y plane defined by a < x<b,csx<d, that contains the point (x), y,) in its interior. If of f(x , y) and — are continuous on R, then there exists an interval I centered at x, and a unique function y(x) defined on I satisfying the initial value problem. y+x +X exi(y- xyrytxa y=" = tx y=% y-xX y-x y#xX +t of 2. that — =M assume tha Ox (x,y) 3. f(x, y) = JM(x,y)dx + g(y) of 4. x s JM(x, y)dx + g'(y) = N(x,y) Section 2.5: = Lincar equation of order one. e ADE. of the form a(x) fy a ayy = 2(x) is said to be a D.E. of order |. e Divide by al(x) we get ey + p(x)y = f(x) x ¢To solve this D.E. we have to find an integrating ~ ¢/P()dx J-p(x)dx feIPCO8 gy + Cel -PCdx The solution is factor. ye Section 2.6 : © Bernouilli’s equation ~ + p(x)y = f(x)y" x Devide by y" => y"dy/dx +p(xy'"= f(x) Let w= y'™ => dw/dx = (1-n)y- ndy/dx > ~ +(1—n)p(x)w = (1- n)f(x) is a linar first order equation. x e Ricatti’s equation ° = p(x) + Q(x)y + R(x)y” d If y, is a known particular solution => y = y,tu and oy =o au dx dx dx =o —(a+2y,R)u= Ru? —-—--— which is Berouilli. x © Claireau equation : y = xy’+f(y’) . =cx+f(c) general solution x=-—f'(t) substituting y = f(t)-tf(t) singular solution. Section 2,7 substitutions. 76 Section 4.1 Let a(x), a\(x), a(x), Toe > a,(x) > 2(X)), 2(X2), stone 28(X,)--.-(A) be continuous on an interval J that contain x,. Let a,(x) # 0 V 13 x, then (A) has a unique solution. ex ix'y” + (-2xy’) + 2y = 6; y(0)=3 yO) = 1 Cx?+x+3 = 0 satisfies the initial value problem but a,(x) = x? =O at x=0 => not a unique solution or it may not be a solution. e Linear dependence : A set of functions f,(x), f(x),----,f,(x) is said to be linearly dependent on an interval | if there exist constans c;, C2, C3, - --c,, not all zeros such that c)f,(x) + c2f,(x) + c3f3+---c, f(x) = 0 on 1. © The set is linearly independent if it is not linearly dependent Orc; =C, =; = ---=c, =0. eA set of functions is linearly dependent on an interval ifat least one . function can be expressed as a linear combination of the remaining functions. ex : f,(x)=Vx+5; f,(x)=V5+5x; £,()=x-1; f(x) =x? are linearly dependent since f,(x) = f,(x) + 5.f;(x) + 0.f,(x) V x in (0, 00) e Theorem 4.2 : For linearly independent funcyions, suppose f;(x), f,(x), --- f,(x) possess at least n-1 derivatives. If the determinant fo"? £,°? f (n-1) n _ for at least one point then the functions are linearly independent. Then determinant = 0 if functions are linearly dependent. e A linear n™ order differencial equation of the form n n-] a, (x) d’y +a,_,(x) d y +--—a, (x) +a,(x)y = 0 (3) is said to be homogeneo: dx". dx" dx * ; n n-1 ifa, (x) d’y +a,_1(x) d y +---a, (x) +a,(x)y = £(x) (4) not homogeneous. dx" dx" dx e Theorem 4.3; Let y,, y2, ---,y, be solutions of the homogeneous linear nth order D.E. (3) on an interval I. Then C1y¥;(x) + coyo(x) + c3y3t---c,y,(x) is also a solution of the homogeneous D.E. 77 Corollaries a) A constant multiple y = c,y,(x) of a solution y,(x) of a homogeneous linear D.E. is also a solution. b) A homogeneous linear D.E. always possesses the trivial solution y = 0. ex : The functions : y,= o*, y,=c™ and y,= ec”, all are solutions of the d’y_.d’y dy 77 s+ dx dx 3 dx => by superposition y = co" +c,e™ c,e™ is a solution. ex : y =x’ isasolution of the homogeneous linear D.L. x’y"-3xy’+4y = 0 => y = cx’ is also a solution ( y = e** and y= 100x”) 6 homogencous equation — 6y = 0 on (-00, +00) © a,)¥1() Haye (yt terFagy = B(%)-(1) a,(x)y" (x) ta@.nQdy” ° +--ta,(x)y =O (2) Let y,, ¥2, ---,¥, be linearly independent solutions of (2) Then y, = cy,(x) + cyy2(x) + c3y3+---c,y,(x) is a general solution of (2) Let y, be any particular solution of (1) Then y = y, ty, is general solution of (1). e Superposition principle for non homogeneous equations. Let Yp1s Ypzs----» Ypx» be the k particular solutions of the linear n” order D.E. (1) on I corresponding to k distinct functions g,(x), g(x), ---, &x(x) Then yp = yp + ¥p2 t----+ ypy is a particular solution of an )¥™ Fan. yXy? +-FaODy = g1(x) + BC) + — + Bd) €X : Yp, = ~4x° is particular solution of y”-3y’+4y = -16x’+14x - 8. Ypo =e” is particular solution of y”-3y’+4y = 2e™ Yp3 = xe" is particular solution of y”-3y’+4y = 2xe* -e* > ¥ =Ypi + Ypo + ¥p3 = -4x"te™+xe" is a solution of y”-3y’+4y = -16x°+24x-8+2°"+2xe"-0" e Any set y;, Y2, ----, ¥, of n linear independent solutions of the © homogeneous linear n' order equation (1) on I is said to be a fundumental set of solutions. Section 4.2 Reduction of 2™ order homogeneous D.E. Let a,(x)y (x) ta,(x)y’ +a,(x)y = 0 (1) be divided by a(x) = y+ p(xy’ + QO@y =0(2)] Let y,(x) be a known solution of (2) Ify=uny (x) = y=uy)"+2y, "uty ,u” y’tpy’+Qy = won" pyr On) + yyu’+(2y,’+pyi)u’= 0 0 . => yyu"t(2yy’+py w= 0 => ywt(2y,’+pyiw= 0 (w= w’) _, Ow, 2% dx + pdx = Inw| + 2 Inly,| =—-Jpdx+C w y1 . Inlwy | =~ [pdx- L wy, = cel" J -pdx ol Pe = w=u=C,— 5 u=C,f 3 dx + C, yi, yi el y=uxdyi(x) = Cy] 2 dx + Cyy1(x) 1 By choosing C, = 0 and C, = 1 a! Pex yo =y¥(X)J—a-4x ¥1