Skip to content

12 Variation of Parameter

Variation of Parameter

for finding particular solution \(y_p\)

more suitable if the RHS function is a \(\log, \tan, \cot, \sec, \csc,\) hyperbolic

  1. Find general solution
\[ y_g = c_1 \textcolor{orange}{y_1} + c_2 \textcolor{orange}{y_2} \]
  1. Let
\[ \begin{aligned} y_p &= v_1 y_1(x) + v_2 y_2(x), \text{where} \\ v_1 &= \int \frac{ \textcolor{orange}{-y_2} \cdot R(x) }{ W(y_1, y_2) } dx \\ v_2 &= \int \frac{ \textcolor{orange}{y_1} \cdot R(x) }{ W(y_1, y_2) } dx \end{aligned} \]

where \(W(y_1, y_2)\) be the Wronskian, then

  1. Complete solution \(y = y_g + y_p\)
Last Updated: 2023-01-25 ; Contributors: AhmedThahir

Comments