Systems of Frst-order differential equations167Figure 4.15.Case 1 (Real distinct roots of the same sign)Here we are considering the general solutionx=c1ertvr+c2estvswhererandsare real and distinct and are either both positive or both negative.We shall assume thatris the larger root in absolute value|r|>|s|. Suppose bothroots are negative, thenr<s<0.Further, suppose the associated eigenvectorsvrandvsare as shown in fgure 4.16 by the heavy arrows. Thus it is quite clear thatast→∞,ert→0 andest→0, and sox→0regardless o± the value o±c1andc2. O± particular signifcance is that i± the initial point lies onvr, thenc2=0 andthe system moves down the line throughvrand approaches the origin over time.
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Initial Point, first-order differential equations, c2 est vs, Real distinct roots