Express the general solution?

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In summary, the general solution is a mathematical formula or equation that satisfies a given differential equation. It is typically expressed using variables and constants, and may involve integration. Finding the general solution allows us to solve the differential equation for any given set of initial conditions, making it important in various fields. The general solution is not always unique, and can be expressed in different forms depending on the type of differential equation and the method used to solve it.

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Express the general solution of x'=(3, 4, -2, -1)x in terms of real-valued functions.

This is 2x2 matrix, 3 and 4 on the left, -2 and -1 on the right. I know that the eigenvalues are 1+2i, 1-2i. And a=1, b=1+i for the first eigenvalue. a=1, b=1-i for the second eigenvalue. But how do I get the answer?

Never mind. I got it.

1. What is the general solution?

The general solution is a mathematical formula or equation that satisfies a given differential equation. It includes all possible solutions to the differential equation, and can be used to find a specific solution for a given set of initial conditions.

2. How do you express the general solution?

The general solution is typically expressed using variables and constants, and may also involve integration. It is important to include all possible solutions and not just one particular solution when expressing the general solution.

3. Why is it important to find the general solution?

Finding the general solution allows us to solve the differential equation for any given set of initial conditions. This is useful in various fields such as physics, engineering, and economics, where differential equations are commonly used to model real-world situations.

4. Is the general solution always unique?

No, the general solution is not always unique. Depending on the type of differential equation, there may be multiple general solutions that satisfy the equation. However, when initial conditions are given, only one specific solution can be found.

5. Can the general solution be expressed in different forms?

Yes, the general solution can be expressed in different forms depending on the type of differential equation and the method used to solve it. For example, a first-order differential equation may have a general solution expressed as a power series, while a second-order differential equation may have a general solution expressed as a linear combination of trigonometric functions.