# Linear homogenous ODEs with constant coefficients

by d.arbitman
Tags: coefficients, constant, homogenous, linear, odes
 P: 103 Given the ODE of the form: y''(x) + A*y'(x) + B*y(x) = 0 If we choose a solution such that y(x) = e$^{mx}$ and plug it into the original ODE, the ODE becomes: (m$^{2}$ + A*m + B)e$^{mx}$ = 0 If we solve for the roots of the characteristic equation such that m = r$_{1}$, r$_{2}$ (root 1 and root 2, respectively) The solution to the ODE would have the form: y(x) = c*e$^{r_{1}*x}$ + d*e$^{r_{2}*x}$, 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$_{1}$ & y$_{2}$, where y$_{1}$ = e$^{r_{1}*x}$ and y$_{2}$ = e$^{r_{2}*x}$ ? 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$^{r_{1}*x}$ + e$^{r_{2}*x}$ + 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.
 HW Helper P: 2,148 That is because the differential equation is linear. That is if L[y]=0 is a linear differential equation and u and v are any two solutions so that L[u]=L[v]=0 then L[a u+a v]=0.
 Math Emeritus Sci Advisor Thanks PF Gold P: 38,879 The fundamental theorem for such equations is: "The set of all solutions to a linear homogeneous differential equation of order n form a vector space of dimension n" That means that if we can find a set of n independent solutions, a basis for that vector space of solutions, any solution can be written as a linear combination of those solutions. And a "linear combination" means a sum of the functions multiplied by constants.

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