3.10 Method of Variation of Parameters

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Last updated 6:55 PM on 6/12/26
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4 Terms

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The Variation of Parameters

Any non-homogenous 2nd order differential equation can be solved by the application of the variation of parameters method; in essence, we must do the following process when applying this algorithm:

  1. Identify the complementary solution of the homogenous version of the function.

  2. Solve for both u1(x) and u2(x)

  3. Multiply both y1(x) and y2(x) by u1(x) and u2(x) respectively

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Solving for u1(x) and u2(x)

y1(x), y2(x) → complementary solution sets

g(x) → forcing function

W(x) → Wronskian of both y1(x) and y2(x)

<p>y<sub>1</sub>(x), y<sub>2</sub>(x) → complementary solution sets</p><p>g(x) → forcing function </p><p>W(x) → Wronskian of both y<sub>1</sub>(x) and y<sub>2</sub>(x) </p>
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Final Solution

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Final Solution Example

Notice how the final solution, y(t), is a combination of both the complementary solution and the product of u1(x) and u2(x).

<p>Notice how the final solution, y(t), is a combination of both the complementary solution and the product of u<sub>1</sub>(x) and u<sub>2</sub>(x). </p>