variation of parameters

Euler equations

Euler equations Here, we consider Euler equations: $$ \tag{1} x^2 y'' + a xy' + by = 0, \quad x\ne 0, $$ where $a$, $b$ are constants. characteristic polynomial Assuming that $y = x^r$ in (1) we obtain $$ \left[r(r-1) + a r + b\right] x^r = 0. $$ Since $x\ne 0$, $x^r\ne 0$, so the equation reduces to $$ \tag{2} r(r-1) + a r + b = 0. $$

Euler equations

Euler equations 這裡我們想要解 Euler equations, 長相如下: $$ \tag{1} x^2 y'' + a xy' + by = 0, $$ 其中 $a$, $b$ 為常數. characteristic polynomial 假設 $y = x^r$, 代入 (1) 可以得到 $$ \left[r(r-1) + a r + b\right] x^r = 0. $$ 假設 $x\ne 0$, 所以 $x^r\ne 0$, 因