Andromeda
Note

Method of Variation of Parameters

Definition

The Method of Variation of Parameters is a general technique to find ypy_p by replacing the constants in yc=c1y1+c2y2y_c = c_1y_1 + c_2y_2 with functions v1(x),v2(x)v_1(x), v_2(x): yp=v1(x)y1(x)+v2(x)y2(x)y_p = v_1(x)y_1(x) + v_2(x)y_2(x)

  • How to read: “The particular solution y subscript p is equal to the function v one of x times y one of x, plus the function v two of x times y two of x.”
  • Meaning: Replace constant coefficients in the complementary solution with unknown functions to build a particular solution.

Why It Matters

Variation of parameters is the ‘universal key’ for nonhomogeneous differential equations; unlike simpler methods, it works for any forcing function, ensuring that engineers can model the response of a system to any real-world input.

Core Concepts

  • The System of Equations: To find v1,v2v_1, v_2, solve:

    1. v1′y1+v2′y2=0v_1'y_1 + v_2'y_2 = 0
    2. v1′y1′+v2′y2′=G(x)/av_1'y_1' + v_2'y_2' = G(x)/a
    • How to read: “The derivative of v one times y one plus the derivative of v two times y two is equal to zero; and the derivative of v one times the derivative of y one plus the derivative of v two times the derivative of y two is equal to G of x divided by a.”
    • Meaning: Two constraints on the derivatives of v1,v2v_1, v_2 that force ypy_p to satisfy the nonhomogeneous ODE.

  • Wronskian Formula: v1′=−y2GaW,v2′=y1GaWv_1' = \frac{-y_2G}{aW}, \quad v_2' = \frac{y_1G}{aW} where W=y1y2′−y2y1′W = y_1y_2' - y_2y_1' is the Wronskian.

    • How to read: “The derivative of v one is equal to negative y two times G, all divided by a times W; the derivative of v two is equal to y one times G, all divided by a times W; where W is the Wronskian, equal to y one times the derivative of y two, minus y two times the derivative of y one.”
    • Meaning / when to use: Closed-form derivatives for the varying parameters; WW in the denominator ensures linear independence of the basis.

  • Integration: The final functions are v1=∫v1′dxv_1 = \int v_1' dx and v2=∫v2′dxv_2 = \int v_2' dx.

Connected Concepts

Connected notes