A Brief Review of Elementary Ordinary Differential …

1 1A Brief Review of Elementary OrdinaryDifferential EquationsAt various points in the material we will be covering, we willneed to recall and use materialnormally covered in an Elementary course on Ordinary Differential equations . In these notes, wewillvery brieflyreview the main topics that will be needed later. For more complete discussionsof these topics, see your current text on Differential equations (by Boyce and DiPrima), or yourold introductory text on Ordinary Differential equations ,or see your instructor s treatment of Basic TerminologyRecall: Adifferential equation(often called a de ) is an equation involving derivatives of anunknown function. If the unknown can be assumed to be a function of only one variable (so thederivatives are the Ordinary derivatives from Calc. I), then we say the Differential equation isan Ordinary Differential equation (ode). Otherwise, the equation is a partial Differential equation(pde).

2 Our interest will just be in odes. In these notes, the variable will usually be denoted byxand the unknown function byyory(x).Recall, also, for any given Ordinary Differential the order of the highest order derivative of the unknown function explicitlyappearing in the any function (or formula for a function) that satisfies solutionis a formula that describes all solutions to the equation. Typically, thegeneral solution to akthorder ode containskarbitrary/undetermined , a Differential equation problem consists of adifferential equation along withsome auxiliary conditions the solution must also satisfy ( , initial values for thesolution). In practice you usually find the general solutionfirst, and then choose valuesfor the undetermined constants so that the auxiliary conditions are online of Elementary Some Analytic Methods for Solving First-OrderODEs(Warning: Here, the word analytic just means that the method leads to exact formulas forsolutions, as opposed to, say, a numerical algorithm that gives good approximations to particularsolutions at fixed points.)

3 Later in this course, the word analytic will mean something else.)Separable equations A first-order ode is separable if it can be written asdydx=g(x)h(y).Such a de can be solved by the following it into the above form ( , the derivative equaling theproduct of a function ofx(theg(x)above), with a function ofy(the aboveh(y)). through byh(y)(but also consider the possibility thath(y)=0 ). both sides with respect tox(don t forget an arbitrary constant). the last equation fory(x).! Example :Consider finding the general solution todydx=2x(y2+1).Going through the above steps:1y2+1dydx=2xH 1y2+1dydxdx= 2x dxH arctan(y)=x2+cH y=tan(x2+c).Linear equations A first-order ode is said to belinearif it can be written in the formdydx+p(x)y=q(x)wherep(x)andq(x)are known functions ofx. Such a Differential equation can be solved bythe following procedure: see, also, chapter 4 of the online text see, also, chapter 5 of the online textversion: 1/3/2013 Review of Elementary it into the above theintegrating factor (x)=e p(x)dx(don t worry about arbitrary constants here).

4 3. (a)Multiply the equation from the first step by the integrating factor.(b)Observe that, by the product rule, the left side of the resulting equation can berewritten asddx[ y], thus giving you the equationddx[ (x)y(x)]= (x)q(x). both sides of your last equation with respect tox, and solve fory(x). Don tforget the arbitrary constant.! Example :Consider finding the general solution toxdydx+4y= through byxgivesdydx+4xy= the integrating factor is (x)=e p(x)dx=e 4/x dx=e4 lnx= the last Differential equation above by this integrating factor and then continuingas described in the procedure:x4[dydx+4xy]=x4[21x2]H x4dydx+4x3y= , by the product rule,ddx[x4y(x)]=x4dydx+4x3y,and so we can rewrite our last Differential equation asddx[x4y(x)]= can be easily integrated and solved: ddx[x4y(x)]dx= 21x6dxH x4y(x)=3x7+cH y(x)=3x7+cx4H y(x)=3x3+cx : 1/3/2013 Review of Elementary ODEs4 Two notes on this formula for the integrating factor (x)is actually derived from the requirement thatddx[ (x)y(x)]= dydx+d dxy= dydx+ py,which is the observation made in step 3b of the procedure.

5 This means that mustsatisfy the simple Differential equationd dx= texts state a formula fory(x)in terms ofp(x)andq(x). The better texts alsostate that memorizing and using this formula is MethodsOther methods for solving first-order Ordinary Differential equations include the integration ofexact equations , and the use of either clever substitutionsor more general integrating factors toreduce difficult equations to either separable, linear orexact equations . See a good de text ifyou are Higher-Order Linear Differential EquationsBasics AnNthorder Differential equation is said to belinearif it can be written in the forma0y(N)+a1y(N 1)+ +aN 2y +aN 1y +aNy=fwherefand theak s are known functions ofx(witha0(x)not being the zero function). Theequation is said to behomogeneousif and only iffis the zero function ( , is always 0 ).Recall that, if the equation is homogeneous, then we have linearity , that is, whenevery1andy2are two solutions to a homogeneous linear Differential equation, andaandbare any twoconstants, theny=ay1+by2is another solution to the Differential equation.

6 In other words,the set of solutions to a homogeneous linear Differential equations is avector spaceof functions.(Isn t it nice to see vector spaces again?)Recall further, general solution to anNthorder linear homogeneous Ordinary Differential equationis given byy(x)=c1y1(x)+c2y2(x)+ +cNyN(x)where theck s are arbitrary constants and{y1,y2, .. ,yN} see, also, chapter 12, sections 1 3, and chapter 14 of the online textversion: 1/3/2013 Review of Elementary ODEs5is a linearly independent set of solutions to the homogeneous de. ( ,{y1,y2, .. ,yN}is a basis for theN-dimensional space of solutions to the homogeneous differentialequation.) general solution to anNthorder linearnonhomogeneous Ordinary Differential equationis given byy(x)=yp(x)+yh(x)whereypis any particular solution to the nonhomogeneous Ordinary Differential equationandyhis a general solution to the corresponding homogeneous real applications,Nis usually 1 or 2.

7 On rare occasions, it may be 4 , and, evenmore rarely, it is 3 . Higher order Differential equations can arise, but usually only in courses ondifferential equations . Do note that ifN=1 , then the Differential equation can be solved usingthe method describe for first order linear equations (see page 2).Notes About Linear IndependenceRecall that a set of functions{y1(x),y2(x), .. ,yN(x)}is linearly independent if and only if none of theyk s can be written as a linear combination ofthe otheryk s . There are several ways to test for linear independence. The one usually discussedin de texts involves the corresponding WronskianW(x), given byW= y1y2y3 yny1 y2 y3 yn y1 y2 y3 yn ..y1(n 1)y2(n 1)y3(n 1) yn(n 1) .The test is that the set ofNsolutions{y1(x),y2(x), .. ,yN(x)}to some givenNth-order homogeneous linear Differential equation is linearly independent if andonly ifW(x0)6=0for any point in the interval over which theseyk s are is a highly recommended test whenN>2 , but, frankly, it is silly to use it whenN=2.

8 Then, we just have a pair of solutions{y1(x),y2(x)}and any such pair is linearly independent if and only if neither function is a constant multiple ofeach other, and THAT is usually obvious upon inspection of the two : 1/3/2013 Review of Elementary ODEs6 Second-Order Linear Homogeneous equations withConstant Coefficients Consider a Differential equation of the formay +by +cy=0wherea,b, andcare (real) constants. To solve such an equation, assume a solution of theformy(x)=erx(whereris a constant to be determined), and then plug this formula foryinto the differentialequation. You will then get the corresponding characteristic equation for the de,ar2+br+c=0 .Solve the characteristic equation. You ll get two values forr,r=r = b b2 4ac2a(with the possibility thatr+=r ). +andr are two distinct real values, then the general solution to the differentialequation isy(x)=c1er+x+c2er xwherec1andc2are arbitrary +=r , thenr+is real and the general solution to the Differential equation isy(x)=c1er+x+c2xer+xwherec1andc2are arbitrary constants.

9 (Note: Thec2xer+xpart of the solution can bederived via the method of reduction of order .) +orr is complex valued, then they are complex conjugates of each other,r+= +i andr = i for some real constants and . The general solution to the Differential equation canthen be written asy(x)=c1e( +i )x+c2e( i )xwherec1andc2are arbitrary constants. However, becausee( i )x=e x[cos( x) isin( x)],the general solution to the Differential equation can also be written asy(x)=C1e xcos( x)+C2e xsin( x)whereC1andC2are arbitrary constants. In practice, the later formula foryis usuallypreferred because it involves just real-valued functions. see, also, chapter 16 of the online textversion: 1/3/2013 Review of Elementary ODEs7! Example :Considery 4y +13y= iny=erx, we getd2dx2[erx] ddx[erx]+13[erx]=0H r2erx 4rerx+13erx=0H r2 4r+13= ,r= ( 4) ( 4)2 4 132=4 362=2 the general solution to the Differential equation can be written asy(x)=c1e(2+3i)x+c2e(2 3i)xor asy(x)=C1e2xcos(3x)+C2e2xsin(3x),with the later formula usually being Euler equations A second-orderEuler equation2is a Differential equation that can be written asax2y +bxy +cy=0wherea,b, andcare (real) constants.

10 To solve such an equation, assume a solution of theformy(x)=xr(whereris a constant to be determined), and then plug this formula foryinto the differentialequation, and solve luck, you will get two distinct real values forr,r1andr2, in which case, the generalsolution to the Differential equation isy(x)=c1xr1+c2xr2wherec1andc2are arbitrary less luck, you only complex values forr, or only one value forr. See chapter 16 ofthe online text to see what to do in these cases.! Example :Considerx2y +xy 9y=0. see, also, chapter 19 of the online text2also called aCauchy-Euler equationversion: 1/3/2013 Review of Elementary ODEs8 Plugging iny=xr, we getx2d2dx2[xr]+xddx[xr] 9[xr]=0H x2[r(r 1)xr 2]+x[rxr 1] 9[xr]=0H r2xr rxr+rxr 9xr=0H r2 9=0H r= the general solution to the Differential equation isy(x)=c1x3+c2x MethodsFor solving more involved homogeneous second-order odes, there is still the method of Frobenius(which we will later discuss in some detail).