D'Alembert Method to solve Diff Eqns

In summary, when using the method of substitution to solve a second order differential equation, we do not need to include any integration constants as they are not necessary in finding the particular solution. This allows us to focus solely on solving for the unknown function u(x) and eliminates any potential issues with having too many unknown parameters in the equation.
  • #1
CAF123
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I understand why this is a good method, but in one of the problems I am trying I yield 4 unknown parameters in a second order differential equation. I believe I should only have 2.

Let f(x) be a part of a homogeneous solution and and u(x) be some unknown function in x. Then a particular solution to the inhomogeneous eqn is y = f(x)u(x).
Subbing this into the differential eqn and cancelling, you get a differential eqn in u(x). Now then solving this will give more integration constants. I think this may be my problem.

Is there a reason why we don't include integration constants at this stage?
 
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  • #2
Yes, there is a reason why we don’t include integration constants at this stage. Integration constants are only needed when solving a differential equation so that the constants can be used to determine the solution to the equation. By using the method of substitution, we are able to reduce the number of unknowns in the equation and focus on finding the particular solution to the equation. This eliminates the need for any integration constants.
 

Related to D'Alembert Method to solve Diff Eqns

1. What is the D'Alembert method for solving differential equations?

The D'Alembert method, also known as the method of characteristics, is a technique used to solve linear partial differential equations. It involves transforming the equation into a set of simpler ordinary differential equations, which can then be solved using standard methods.

2. How does the D'Alembert method work?

The D'Alembert method works by transforming the given partial differential equation into a system of ordinary differential equations. This is done by introducing new variables and using the characteristics of the equation to eliminate the partial derivatives. The resulting system of equations can then be solved using standard methods, such as separation of variables or variation of parameters.

3. When is the D'Alembert method useful?

The D'Alembert method is useful for solving linear partial differential equations with constant coefficients. It is particularly useful for solving wave equations, such as the heat equation and the wave equation, as it allows for the separation of variables and the use of Fourier series techniques.

4. What are the limitations of the D'Alembert method?

The D'Alembert method is only applicable to linear partial differential equations with constant coefficients. It is also limited to homogeneous equations, meaning that the right-hand side of the equation must equal zero. Additionally, the method may not work for equations with non-constant coefficients or non-linear terms.

5. Are there any alternative methods to solve differential equations?

Yes, there are many alternative methods for solving differential equations, such as the method of undetermined coefficients, variation of parameters, and Laplace transforms. The choice of method depends on the type of differential equation and its properties, as well as personal preference and experience of the solver.

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