Register to reply

Linear homogenous ODEs with constant coefficients

by d.arbitman
Tags: coefficients, constant, homogenous, linear, odes
Share this thread:
d.arbitman
#1
Nov12-12, 01:39 PM
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[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.
Phys.Org News Partner Science news on Phys.org
Physicists discuss quantum pigeonhole principle
Giant crater in Russia's far north sparks mystery
NASA Mars spacecraft prepare for close comet flyby
lurflurf
#2
Nov12-12, 07:27 PM
HW Helper
P: 2,263
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.
HallsofIvy
#3
Nov13-12, 07:58 AM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,338
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.


Register to reply

Related Discussions
Analytically solving ODEs with non-constant coefficients for a specific t Differential Equations 1
Homogenous constant coefficient linear differential equations Differential Equations 1
Systems of Linear Homogenous Differential equations with Constant Coefficients Differential Equations 1
Uniqueness Theorem for homogenous linear ODEs Linear & Abstract Algebra 5