# Is this behaviour generalizable?

• Whovian
In summary, the classic linearly independent solutions for second-order linear homogeneous constant-coefficient ODEs are ##y_1=e^{r_1\cdot t}## and ##y_2=e^{r_2\cdot t}##, and for Cauchy-Euler equations are ##y_1=t^{r_1}## and ##y_2=t^{r_2}##. If these solutions are linearly dependent, then the independent solutions chosen are ##y_1=e^{r\cdot t}## and ##y_2=t\cdot y_1=\frac{\partial y_1}{\partial r}## for constant-coefficient equations, and ##y_1=t^r
Whovian
Notice that, when solving second-order linear homogeneous constant-coefficient ODEs, the classic linearly independent solutions are ##y_1=e^{r_1\cdot t}## and ##y_2=e^{r_2\cdot t}## for roots ##r_1,r_2## of our auxiliary equation. (Of course, we can pick any other basis for this two-dimensional vector space.)

If those are linearly dependent, the independent solutions chosen are ##y_1=e^{r\cdot t}## (no surprise) and ##y_2=t\cdot y_1=\frac{\partial y_1}{\partial r}##.

Similarly, for homogeneous second-order (insert long list of adjectives here) Cauchy-Euler equations, we set up a quadratic equation and find ##y_1=t^{r_1}## and ##y_2=t^{r_2}## as our linearly independent solutions, or, if those are linearly dependent, ##y_1=t^r## and ##y_2=t^r\cdot\log\left(t\right)=\frac{\partial y_1}{\partial r}##.

This is suggesting that, for nth-order linear homogeneous ODEs, if we can guess solutions of the form ##f\left(r,t\right)##,

If n solutions ##r_1,\ldots,r_n## show up for ##r##, we can use ##\left\{f\left(r_1,t\right),\ldots,f\left(r_n,t\right)\right\}## as a basis for the set of solutions.

If any repeated roots show up (slightly sketchy since it might not be a polynomial in terms of ##r##,) we use ##f\left(r_1,t\right),\ldots,f\left(r_n,t\right)## along with the partial derivatives with respect to ##r_k##, up to the ##\mathrm{mult}\left(r_k\right)##th derivative, for all ##k##, as our basis.

Now, for obvious reasons, this probably only holds in specific situations. Have I run into the specific couple for which our linearly independent solutions are generated by taking the partial derivative if a repeated root shows up, or is this generalizable in any way such as the near-unreadable example I suggested?

How generalizable?
Cauchy-Euler equations are related to constant-coefficient by the chain rule.
$$\left( \dfrac{dx}{du} \dfrac{d}{dx} \right)^n=\left( \dfrac{d}{du} \right)^n$$
so if u=g(x)
$$\mathrm{y}_2(u)=u \, \mathrm{y}_1(u)$$
then
$$\mathrm{y}_2(\mathrm{g}(x))=\mathrm{g}(x) \, \mathrm{y}_1(\mathrm{g}(x))$$
u=log(x) for Cauchy-Euler equations
u=x for constant-coefficient

## 1. What does it mean for behavior to be "generalizable"?

Generalizability in behavior refers to the ability to apply the findings from a study or experiment to a wider population or situation. It is a measure of how well the results of a study can be generalized to people or situations outside of the specific sample or context used in the study.

## 2. How do you determine if behavior is generalizable?

The generalizability of behavior can be determined by looking at the characteristics of the study sample and comparing them to the characteristics of the larger population. If the sample is representative of the population in terms of demographics, behaviors, and other relevant factors, then the behavior may be considered generalizable.

## 3. What factors can affect the generalizability of behavior?

There are several factors that can affect the generalizability of behavior, including the sample size, sample selection methods, study design, and cultural or environmental factors. It is important to consider these factors when interpreting the results of a study.

## 4. Can behavior be generalizable to all populations and situations?

No, behavior may not be generalizable to all populations and situations. Every study has limitations and it is important to consider the specific context in which the behavior was observed. Factors such as cultural differences, individual differences, and unique circumstances can impact the generalizability of behavior.

## 5. How can researchers increase the generalizability of their findings?

Researchers can increase the generalizability of their findings by using a larger and more diverse sample, carefully selecting participants to ensure they are representative of the larger population, and using multiple methods and measures to collect data. Additionally, conducting studies in different settings and replicating the study with different populations can also help increase the generalizability of behavior.

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