# Euler substitution

When given a differential equation of the form:

ay'' + by' + cy = 0,

you can utilize Euler substitution by assuming y = eλt. This yields:

y = eλt y' = λeλt y'' = λ2eλt

Substituting back in, this yields:

aλ2eλt + bλeλt + ceλt = 0

Dividing through by eλt,

aλ2 + bλ + 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 $\sqrt {b^2-4ac} < 0$,

λ1 = r + iμ and λ2 = riμ,

y = c1y1 + c2y2 = ert(c1cosμt + c2sinμt)

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

y = c1y1 + c2y2 = eλt(c1 + c2t)