I understand that to solve an equation such as:
$$y''+5y'+4y=0,$$
y: y(x)
I must consider $D=\frac{d}{dx}$ and the auxiliary equation:
$$(D^2+5D+4)y=0,$$
or the separate, separable equations:
$$(D+1)y=0, (D+4)y=0.$$
From this, it is clear that solutions are:
$$y=c_1 e^{-4x}+c_2 e^{-x}.$$
First of all, I was wondering if anyone could offer me any intuition of why we can combine the two solutions (simply by addition) to get a general solution.
But the main reason for the post, was because when considering an auxiliary equation that has equal roots, I have read that we cannot simply write:
$$y=c_1e^{ax}+c_2e^{ax}.$$
and for:
$$(D-a)(D-a)y=0.$$
We must let:
$$u=(D-a)y.$$
solving gives:
$$u=Ae^{ax}$$
Which by substitution into the auxiliary equation gives a clear solution:
$$y=(Ax+B)e^{ax}.$$
I was just wondering why the first solution is not valid in the case when roots are equal, also I am not sure what "u" is/ where it came from.
Any help is appreciated, Thanks!
Best Answer
The general solution of a linear homogeneous ODE of $n$th order is given as a linear combination $$y=c_1y_1+c_2y_2+\ldots+c_ny_n$$ where $y_1,\,y_2,\,\ldots,\,y_n$ are $n$ linearly independent solutions of the ODE and $c_1,\,c_2,\,\ldots,\,c_n$ are arbitrary constants. For a second order equation, two equal roots of the auxiliar equation give only one solution, since $$y=c_1e^{at}+c_2e^{at}=(c_1+c_2)e^{at}$$ so, we need a second solution $y_2$, such that $e^{at}$ and $y_2$ were linearly independent.