MA 36600 FINAL REVIEWChapter 7§7.1: Introduction.•Asystem of first order equationsis a collection of initial value problemsdxkdt=Gk(t, x1, x2, . . . , xn),xk(t0) =x0k;k= 1,2, . . . , m.We say that such a system is asystem of first order linear equationsif we can writeGk(t, x1, x2, . . . , xn)=pk1(t)x1+pk2(t)x2+· · ·+pkn(t)xn+gk(t);k= 1,2, . . . , m.If at least one of theGk(t, x1, x2, . . . , xn) is not in the form above, we call the systemnonlinear. Wecall a linear systemhomogeneousif eachgk(t) is the zero function. Otherwise, we call the systemnonhomogeneous.•We can always express annth order linear equation as a system of first order equations: In general,annth order equationis an ordinary differential equation in the formy(n)=Gt, y, y(1), . . . , y(n−1)where the notationy(k)denotes thekth derivative; and we have the initial conditionsy(t0) =y0,y(1)(t0) =y(1)0,. . .y(n−1)(t0) =y(n−1)0.If we make the substitutionsx1=yx2=y(1)...xn=y(n−1)=⇒x1=x2...xn−1=xnxn=G(t, x1, . . . , xn)wherex01=y0x02=y(1)0...x0n=y(n−1)0•Assume that there are as many equations as variables i.e., that
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Elementary algebra, Xn, First Order Equations, nonlinear systems, Order Linear Equations