Variation of parameters

If \(y_1,y_2\) form a fundamental set of \(y''+py'+qy=0\), the idea is to replace the constants in \(c_1y_1+c_2y_2\) by functions: \(y_p=u_1y_1+u_2y_2\). Since there are two unknown functions, one also imposes \(u_1'y_1+u_2'y_2=0\), and substituting into the equation gives \(u_1'y_1'+u_2'y_2'=g\). This is a \(2\times2\) system whose determinant is the Wronskian \(W\neq0\); by Cramer, \(u_1'=-y_2g/W\) and \(u_2'=y_1g/W\), and integrating yields the formula above.

The equation must be in standard form, with coefficient \(1\) on \(y''\): in \(2y''+2y=8t\) divide first, so \(g=4t\), not \(8t\). The constants of integration are omitted, because they would only add homogeneous solutions. With \(y''+y=\sec t\), \(y_1=\cos t\), \(y_2=\sin t\) and \(W=1\), one gets \(u_1=\ln\cos t\) and \(u_2=t\), so \(y_p=\cos t\ln\cos t+t\sin t\); undetermined coefficients would not have worked here.

The costs are that \(y_1,y_2\) are needed beforehand (for example from the characteristic equation) and the integrals can be hard; when \(g\) is a polynomial, exponential or trigonometric function, undetermined coefficients is usually faster. The general solution is always \(y=c_1y_1+c_2y_2+y_p\). The method also extends to order \(n\) with an \(n\times n\) system whose determinant is the Wronskian of \(n\) functions.