\(y'' + py' + qy = 0 ~~~ y = e^{rx}\)
\(将 ~~ y = e^{rx} ~~ 代入 ~~ y'' + py' + qy = 0 ~~~ y = e^{rx}\)
\(得 ~~ (r^{2} + pr + q)e^{rx} = 0\)
\(即\)
\(r^{2} + pr + q = 0\)
\(解出r的两根r_{1},r_{2}\)
\(按下表即可求出微分方程的通解\)
| \(特征方程r^{2} + pr + q = 0的两根r_{1},r_{2}\) | \(微分方程y'' + py' + qy = 0的通解\) |
|---|---|
| \(r_{1} \ne r_{2}\) | \(y = C_{1}e^{r_{1}x} + C_{2}e^{r_{2}x}\) |
| \(r_{1} = r_{2} = r\) | \(y = (C_{1} + C_{2}x)e^{rx}\) |
| \(r_{1} = \alpha + i \beta , r_{2} = \alpha-i\beta\) | \(y = e^{\alpha x}(C_{1}cos\beta x + C_{2}sin \beta x)\) |