Variation of parameters Finding a particular solution when a homogeneous solution appears in the right-side function is handled using a method called variation of parameters. (This method is also used to find particular solutions of equations that cannot be handled by undetermined coefficients.) The following steps show how variation of parameters is used to find the particular solution of one specific equation.
a. Consider the equation y" – y – et. Show that the homogeneous solutions are y1 = et and y2 = e–t. Note that the right–side function is a homogeneous solution.
b. Assume a particular solution has the form
where the functions u1 and u2 are to be determined. Compute yp' and impose the condition u1'et + u2'e–t = 0 to show that ypʹ = u1et – u2′e–t.
c. Compute yp" and substitute it into the
d. Parts (c) and (d) give two equations for u1ʹ and u2ʹ Eliminate u2ʹ and show that the equation for u1 is
e. Solve the equation in part (d) for u1.
f. Use part (e) to show that the equation for u2 is
g. Solve the equation in part (f) for u2.
h. Now assemble the particular solution yp(t) =u1(t)et + u2(t)e–t and show that
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Calculus: Early Transcendentals, 2nd Edition
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage