Given the ODE of the form:(adsbygoogle = window.adsbygoogle || []).push({});

y''(x) + A*y'(x) + B*y(x) = 0

If we choose a solution such that y(x) = e[itex]^{mx}[/itex]

and plug it into the original ODE, the ODE becomes:

(m[itex]^{2}[/itex] + A*m + B)e[itex]^{mx}[/itex] = 0

If we solve for the roots of the characteristic equation such that

m = r[itex]_{1}[/itex], r[itex]_{2}[/itex] (root 1 and root 2, respectively)

The solution to the ODE would have the form:

y(x) = c*e[itex]^{r_{1}*x}[/itex] + d*e[itex]^{r_{2}*x}[/itex], where c and d are constants

My question is, why are the constants where they are in the solution? In other words, why are they multiplying y[itex]_{1}[/itex] & y[itex]_{2}[/itex], where

y[itex]_{1}[/itex] = e[itex]^{r_{1}*x}[/itex] and y[itex]_{2}[/itex] = e[itex]^{r_{2}*x}[/itex] ?

Why are the constants not just added to y(x), such that the solution to the ODE would be as follows, where k is a constant.

y(x) = e[itex]^{r_{1}*x}[/itex] + e[itex]^{r_{2}*x}[/itex] + k

This question mainly is for second order and higher differential equations. I understand how it works for first order linear homogenous DEs because the constant is simply the constant of integration, but I am having trouble understanding how it applies to higher orders.

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# Linear homogenous ODEs with constant coefficients

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