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Method of Variation of Parameters

In a Nut Shell: Recall that the method of undetermined coefficients to solve for a

particular solution of a nonhomogeneous d.e. of the form

a d²y/dx² + b dy/dx + cy = f(x)

Variation of parameters, gives an alternative way to solve for particular solutions for general

types of functions, f(x), beyond those covered using the method of undetermined coefficients.

Strategy: The method of variation of parameters starts with calculating the complementary

solution, y_c, of the d.e. d²y/dx² + b dy/dx + cy = 0

y_c = C₁ y₁(x) + C₂ y₂(x)

where y₁(x) and y₂(x) are two linearly independent solutions to the homogeneous

d.e. In variation of parameters form the particular solution by using two new functions,

u₁(x) and u₂(x), yet to be determined, as follows:

y_p(x) = u₁(x) y₁(x) + u₂(x) y₂(x)

Continue by calculating the first derivative of y_p(x) as follows:

dy_p(x)/dx = du₁(x)/dx y₁(x) + du₂(x)/dx y₂(x) + u₁(x) dy₁(x)/dx + u₂(x) dy₂(x)/dx

Next comes a key step: To avoid second derivatives of u₁ and u₂ set

du₁(x)/dx y₁(x) + du₂(x)/dx y₂(x) = 0 (equation 1)

leaving dy_p(x)/dx = u₁(x) dy₁(x)/dx + u₂(x) dy₂(x)/dx take another derivative

d²y_p(x)/dx² = du₁(x)/dx dy₁(x)/dx + du₂(x) dy₂(x)/dx

+ u₁(x) d²y₁(x)/dx² + u₂(x) d²y₂(x)/dx²

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