Chapter 5Thus, a general solution to the given system isu(t)=c1−12c2e−t+12et+53t,v(tc1+c2e−t+53t.9.Expressed in operator notation, this system becomes(D+2)[x]+D[y]=0,(D−1)[x]+(D−1)[y]=sint.In order to eliminate the functiony(t), we will apply the operator (D−1) to the Frst equationabove and the operator−Dto the second one. Thus, we have(D−1)(DxD−1)D[y]=(D−1) = 0,−D(D−1)[x]−D(D−1)[y]=−D[sint−cost.Adding these two equations yields the di±erential equation involving the single functionx(t)given by±(D2+D−2)−(D2−D)²[x−cost⇒2(D−1)[x−cost.(5.3)This is a linear Frst order di±erential equation with constant coeﬃcients and so can be solvedby the methods of Chapter 2.(See Section 2.3.)However, we will use the methods of
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This note was uploaded on 03/29/2010 for the course MATH 54257 taught by Professor Hald,oh during the Spring '10 term at University of California, Berkeley.