Consider the equationy 3y 4y = 2et (i)from Example 5. Recall that y1(t) = et and y2(t) =

Chapter 3, Problem 29

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Consider the equationy 3y 4y = 2et (i)from Example 5. Recall that y1(t) = et and y2(t) = e4t are solutions of the correspondinghomogeneous equation. Adapting the method of reduction of order (Section 3.4), seek asolution of the nonhomogeneous equation of the form Y(t) = v(t)y1(t) = v(t)et, wherev(t) is to be determined.(a) Substitute Y(t), Y(t), and Y(t) into Eq. (i) and show that v(t) must satisfyv 5v = 2.(b) Let w(t) = v(t) and show that w(t) must satisfy w 5w = 2. Solve this equationfor w(t).(c) Integrate w(t) to find v(t) and then show thatY(t) = 25 tet + 15 c1e4t + c2et.The first term on the right side is the desired particular solution of the nonhomogeneousequation. Note that it is a product of t and et.