# Euler substitution

### From Conservapedia

When given a differential equation of the form:

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

you can utilize **Euler substitution** by assuming *y* = *e*^{λt}. This yields:

*y* = *e*^{λt}
*y*' = λ*e*^{λt}
*y*'' = λ^{2}*e*^{λt}

Substituting back in, this yields:

*a*λ^{2}*e*^{λt} + *b*λ*e*^{λt} + *c**e*^{λt} = 0

Dividing through by *e*^{λt},

*a*λ^{2} + *b*λ + *c* = 0

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

**Case I: When ,**

**Case II: When ,**

λ_{1} = *r* + *i*μ and λ_{2} = *r* − *i*μ,

*y* = *c*_{1}*y*_{1} + *c*_{2}*y*_{2} = *e*^{rt}(*c*_{1}*c**o**s*μ*t* + *c*_{2}*s**i**n*μ*t*)

**When Case III: ,**

*y* = *c*_{1}*y*_{1} + *c*_{2}*y*_{2} = *e*^{λt}(*c*_{1} + *c*_{2}*t*)