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Trying2Learn

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- TL;DR Summary
- Why do we expect n solutions to an n-th order differential equation

For a linear

Fine...

By finding n linearly independent solutions, we are essentially covering all the necessary components to form the general solution. These solutions form a basis for the solution space, enabling us to express any solution to the differential equation as a linear combination of them. This approach ensures that you capture all the possible behaviors and variations allowed by the equation.

Fine... I get that.

For example, if it is a second order, we have two remaining "unknowns": the first derivative and the zero derivative (the equation itself).

Thus, it seems "justified" in searching for two.

However, consider the case, in a second order (spring, mass, damping), when the roots of the characteristic, repeat. We find a second solution by multiplying the first, by say, time. No problem.

However, my question is that I do NOT see this "search" for a second solution, as motivated by finding expressions for the first or zeroth order.

The process just seems whimsical.

I suppose I am stuck between two issues: 1) why must there be as many solutions as order of the equation and 2) is the search for other solutions grounded in the exploration of "n" solutions (all the the n-1, n-2, n-3, 0) terms?

In fact, to go further, when we do get the quadratic characteristic equation for an oscillator, the act of claiming two solutions seems serendipitous, rather than grounded in a search for a solution for position, and a second for velocity.

I am not sure I am asking this the right way.

Maybe I should be asking: why do the n solutions form a "basis" and how do we know that basis is of the same order of the differential equation, and why do such searches often seem infused with "guess work?"

**th order differential equation with constant coefficients, the general solution can be expressed as a linear combination of***n***linearly independent solutions.***n*Fine...

By finding n linearly independent solutions, we are essentially covering all the necessary components to form the general solution. These solutions form a basis for the solution space, enabling us to express any solution to the differential equation as a linear combination of them. This approach ensures that you capture all the possible behaviors and variations allowed by the equation.

Fine... I get that.

For example, if it is a second order, we have two remaining "unknowns": the first derivative and the zero derivative (the equation itself).

Thus, it seems "justified" in searching for two.

However, consider the case, in a second order (spring, mass, damping), when the roots of the characteristic, repeat. We find a second solution by multiplying the first, by say, time. No problem.

However, my question is that I do NOT see this "search" for a second solution, as motivated by finding expressions for the first or zeroth order.

The process just seems whimsical.

I suppose I am stuck between two issues: 1) why must there be as many solutions as order of the equation and 2) is the search for other solutions grounded in the exploration of "n" solutions (all the the n-1, n-2, n-3, 0) terms?

In fact, to go further, when we do get the quadratic characteristic equation for an oscillator, the act of claiming two solutions seems serendipitous, rather than grounded in a search for a solution for position, and a second for velocity.

I am not sure I am asking this the right way.

Maybe I should be asking: why do the n solutions form a "basis" and how do we know that basis is of the same order of the differential equation, and why do such searches often seem infused with "guess work?"

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