Solving IVP without Initial Equation: Step-by-Step Guide

To create an initial equation, you need a slope at that point, which you don't have. However, you can still solve for x1(t) and x2(t) separately by plugging in the given initial values and using integration to find their respective equations. This may not be the most efficient or accurate method, but it is one approach to solving this type of IVP.
  • #1
cue928
130
0
Hey, I need some guidance on an IVP. In general, how do you proceed on these types of problems when you have only the initial values but no initial equation? For example, I have
x1(0)=1 and x2(0)=0 but that is it. I understand, for example, how to do IVP's in the context of separating variables, exact, laplace, etc, but I have no clue how to proceed on this.
 
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  • #2
cue928 said:
Hey, I need some guidance on an IVP. In general, how do you proceed on these types of problems when you have only the initial values but no initial equation? For example, I have
x1(0)=1 and x2(0)=0 but that is it. I understand, for example, how to do IVP's in the context of separating variables, exact, laplace, etc, but I have no clue how to proceed on this.
That's really all you are given? If so, what you have is a single point in the x1 - x2 plane at (1, 0).
 

1. How do I solve an IVP without an initial equation?

To solve an IVP without an initial equation, you will need to use the method of undetermined coefficients. This involves finding a particular solution and then adding it to the homogeneous solution to get the general solution.

2. What is the method of undetermined coefficients?

The method of undetermined coefficients is a technique used to solve differential equations with constant coefficients. It involves finding a particular solution that satisfies the equation and then adding it to the homogeneous solution to get the general solution.

3. Can I use the method of undetermined coefficients for any type of IVP?

Yes, the method of undetermined coefficients can be used for any type of IVP as long as the equation has constant coefficients. This method is particularly useful for solving second-order linear differential equations.

4. How do I find the particular solution using the method of undetermined coefficients?

To find the particular solution, you will need to follow a systematic process of guessing a solution based on the form of the non-homogeneous part of the equation. This solution is then substituted into the original equation to determine the coefficients.

5. What is the homogeneous solution in the method of undetermined coefficients?

The homogeneous solution is the solution to the equation without the non-homogeneous part. It is usually found by assuming a solution in the form of e^rt, where r is a constant. This solution is then added to the particular solution to get the general solution.

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