Consider the equation y?? ? 3y? ? 4y = 2e?t (i) from Exa
Question and Solution
Consider the equation y?? ? 3y? ? 4y = 2e?t (i) from Example 5. Recall that y1(t) = e?t and y2(t) = e4t are solutions of
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Consider the equation y?? ? 3y? ? 4y = 2e?t (i) from Example 5. Recall that y1(t) = e?t and y2(t) = e4t are solutions of the corresponding homogeneous equation. Adapting the method of reduction of order (see the discussion preceding Problem 28 in Section 4.2), seek a solution of the nonhomogeneous equation of the form Y(t) = ?(t)y1(t) = ?(t)e?t , where ?(t) is to be determined. (a) Substitute Y(t), Y? (t), and Y??(t) into Eq. (i) and show that ?(t) must satisfy ??? ? 5?? = 2. (b) Let w(t) = ?? (t) and show that w(t) must satisfy w? ? 5w = 2. Solve this equation for w(t). (c) Integrate w(t) to find ?(t) and then show that Y(t) = ?2 5 te?t + 1 5 c1e4t + c2e?t . The first term on the right side is the desired particular solution of the nonhomogeneous equation. Note that it is a product of t and e?t . Nonhomogeneous Cauchy
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