General Solution for a 2x2 Matrix Differential Equation

In summary, a general solution is a mathematical expression that represents all possible solutions to a given equation or problem. It differs from a particular solution in that it includes arbitrary constants or parameters that allow for a range of solutions. An example of expressing a general solution is y = x^2 + C for the differential equation dy/dx = 2x, where C is an arbitrary constant. It is important to express the general solution in order to find all possible solutions and gain a comprehensive understanding of the problem. However, there can be limitations to expressing a general solution and it is important to verify its validity for the given problem.
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Express the general solution of x'=(2, 9/5, -5/2, -1)x in terms of real-valued functions.

(this is 2x2 matrix, 2 and 9/5 on the left, -5/2 and -1 on the right. The complex roots are (1/2)+(3/2)i and (1/2)-(3/2)i and a=1, b=(3/5)+(3/5)i for the first root. And a=1, b=(3/5)-(3/5)i for the second root. But I don't know how to get the answer.
 
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Never mind. I solved it.
 

1. What is a general solution?

A general solution is a mathematical expression that represents all possible solutions to a given equation or problem. It includes any arbitrary constants or parameters that allow for a range of solutions.

2. How is a general solution different from a particular solution?

A particular solution is a specific solution to a given problem, while a general solution is a formula that represents all possible solutions. A particular solution will have specific values for all variables, while a general solution will have arbitrary constants that can take on different values.

3. Can you provide an example of expressing a general solution?

Sure, for the differential equation dy/dx = 2x, the general solution would be y = x^2 + C, where C is an arbitrary constant. This general solution can represent all possible solutions to the given differential equation.

4. Why is it important to express the general solution?

Expressing the general solution allows us to find all possible solutions to a given problem or equation. It provides a comprehensive understanding of the problem and allows for further analysis and manipulation of the solution.

5. Are there any limitations to expressing a general solution?

Yes, there can be limitations to expressing a general solution. In some cases, the general solution may not be able to represent all possible solutions or may not be applicable for certain values of the variables. It is important to check the validity of the general solution for the given problem.

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