I am taking a course in Differential Equations and we were shown how to use the auxiliary equation to easily get the general solution for a differential equations with constant coefficients. For example:
$$
y'' – 4y' + 16y = 0
$$
has the auxiliary equation:
$$
m^2 – 4 + 16= 0
$$
However, I wanted to know how one can write the auxiliary equation for a DE with undetermined coefficients. For example:
$$
2x^2y'' + 5xy' + y = 0
$$
Thank you!
Best Answer
To clarify the question about linearity :
Both equations
$y'' - 4y' + 16y = 0$
$2x^2y'' + 5xy' + y = 0$
are linear.
For ODEs, linear means linear relatively to $y$ , $y'$, $y''$ even if the coefficients of them are not linear.
To solve $\quad y'' - 4y' + 16y = 0\quad$ change of function : $$y(x)=c\:e^{m\:x}$$ in order to obtain $$m^2-4m+16=0$$
To solve $\quad 2x^2y'' + 5xy' + y = 0\quad$ change of function : $$y(x)=c\:x^m $$ in order to obtain $$2m(m-1)+5m+1=0$$