Euler substitution

When given a differential equation of the form:

$a y'' + b y' + c y = 0$ ,

you can utilize Euler substitution by assuming $y = e λ t$ . This yields:

$y = e λ t$ $y' = λ e λ t$ $y'' = λ 2 e λ t$

Substituting back in, this yields:

$a λ 2 e λ t + b λ e λ t + c e λ t = 0$

Dividing through by $e λ t$ ,

$a λ 2 + b λ + c = 0$

Then, perform the quadratic formula. There are three cases that arise:

Case I: When $\sqrt {b^2-4ac} > 0$ ,

$y = c_1 y_1 + c_2 y_2 = c_1 e^{\lambda_1 t} + c_2 e^{\lambda_2 t}$

Case II: When $\sqrt {b^2-4ac} < 0$ ,

$λ 1 = r + i μ$ and $λ 2 = r - i μ$ ,

$y = c 1 y 1 + c 2 y 2 = e r t (c 1 c o s μ t + c 2 s i n μ t)$

When Case III: $\sqrt {b^2-4ac} = 0$ ,

$y = c 1 y 1 + c 2 y 2 = e λ t (c 1 + c 2 t)$