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HallsofIvy said:For this differential equation, the characteristic equation is [itex]r^2= -\lambda_i[/itex] and has characteristic roots [itex]r= \pm\lambda_i i[/itex] (that second "i" is the imaginary unit). That, in turn means that the general solution to the differential equation is [itex]\psi_i(x)= Ae^{i\lambda_i x}+ Ce^{-i\lambda_i x}= Ccos(\lambda_i x)+ D sin(\lambda_i x)[/itex] where A, B, C, and D are constants. Do you understand how to convert from the complex exponential to sine and cosine? It is based on [itex]e^{ix}= cos(x)+ i sin(x)[/itex].
Cyosis said:That is all correct so far. You can now solve equation (5) for C3 and then plug it into solve for C4. After that you can solve C1 and C2. It doesn't look like it's going to be pretty though, I'll have a look at it later to see if you can simplify it.
We don't. If [itex]cosh(\mu)cos(\mu)= 1[/itex], so there are non-trivial solutions, there will be an infinite number of such non-trivial functions.ssky said:thanks alot,
if [itex]C_1\ne 0[/itex]
then how we find [itex]C_1[/itex] ?
[/URL]ssky said:thank you very much... I would like to give you this gift for helping me
http://blog.doctissimo.fr/php/blog/un_avenir_heureux/images/bouquet%20de%20fleur.gif
A differential equation is a mathematical equation that relates a function with its derivatives. It describes the relationship between a function and its rate of change over time or space.
The solution of a differential equation is the function that satisfies the equation when substituted into it. It represents the relationship between the variables in the equation and can be used to predict the behavior of the system described by the equation.
The process of solving a differential equation involves finding the function that satisfies the equation by using various mathematical techniques such as separation of variables, substitution, or integration.
There are two types of solutions to a differential equation - explicit and implicit. An explicit solution is a function expressed in terms of the independent variable, while an implicit solution is a relationship between the independent and dependent variables.
Differential equations are essential in many fields of science and engineering, such as physics, chemistry, biology, and economics. They are used to model real-world systems and make predictions about their behavior. They also provide a powerful tool for understanding and describing natural phenomena.