Linear homogenous ODEs with constant coefficients

In summary: So in the case of the ODE above, if we want to find y(x) as a linear combination of u(x) and v(x), we would sum them both up and get y(x) = c*u(x) + d*v(x).
  • #1
d.arbitman
101
4
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.
 
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  • #2
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=L[v]=0 then L[a u+a v]=0.
 
  • #3
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.
 

1. What is a linear homogenous ODE with constant coefficients?

A linear homogeneous ODE with constant coefficients is a type of ordinary differential equation (ODE) that can be written in the form: y'(x) + a0y(x) = 0, where a0 is a constant. This means that the derivative of the dependent variable, y(x), is directly proportional to the function itself, and there are no other terms involving the independent variable, x.

2. How do you solve a linear homogenous ODE with constant coefficients?

To solve a linear homogenous ODE with constant coefficients, you can use the method of undetermined coefficients or the method of variation of parameters. Both methods involve finding a general solution by assuming a form for the solution and then solving for the unknown coefficients. The method of undetermined coefficients is more straightforward but can only be used for certain types of ODEs, while the method of variation of parameters can be used for any linear ODE with constant coefficients.

3. What is the difference between a linear homogenous ODE and a non-homogenous ODE?

A linear homogenous ODE has all terms containing the dependent variable, y, and its derivatives, while a non-homogenous ODE has additional terms that involve the independent variable, x. This means that the solution to a linear homogenous ODE will be a linear combination of exponential functions, while the solution to a non-homogenous ODE will also include a particular solution that satisfies the additional terms.

4. When are linear homogenous ODEs with constant coefficients commonly encountered?

Linear homogenous ODEs with constant coefficients are commonly encountered in various fields of science and engineering, including physics, chemistry, biology, and economics. They are used to model systems where the rate of change of a variable is dependent on the current value of that variable, such as in population growth or radioactive decay.

5. What are the applications of solving linear homogenous ODEs with constant coefficients?

The ability to solve linear homogenous ODEs with constant coefficients is important in many areas of science and engineering. It allows us to accurately model and predict the behavior of systems that exhibit exponential growth or decay, as well as systems that involve oscillations or vibrations. It is also a fundamental tool in understanding and analyzing complex phenomena, such as fluid flow, heat transfer, and electrical circuits.

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