Exercise · ZW-VARIATION-OF-PARAMETERS-004 · ★☆☆ Basic
When computing \(u_1\) and \(u_2\) by integration, why can the constants of integration be omitted?
Choose one option.
A They only add solutions of the homogeneous equation, already in \(c_1y_1+c_2y_2\) B They change the equation being solved C They must be fixed by the initial conditions before substituting D They make \(W\) stop being nonzero Show answer A. They only add solutions of the homogeneous equation, already in \(c_1y_1+c_2y_2\)← Back to the question
Explanation A constant in \(u_1\) or \(u_2\) adds a multiple of \(y_1\) or \(y_2\) to \(y_p\), which is still a solution of the homogeneous equation; it is absorbed in \(c_1,c_2\) of the general solution .
A method that builds a particular solution \(y_p\) of \(y''+py'+qy=g(t)\) from two homogeneous solutions \(y_1,y_2\) and their Wronskian \(W\), by letting the constants vary.
\[y_p=-y_1\!\int\!\frac{y_2\,g}{W}\,dt+y_2\!\int\!\frac{y_1\,g}{W}\,dt\]
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