16.1 Show that e^{2x}, sin(2x) is linearly independent on + infinity -infinity

In summary, the conversation discusses how to show that $e^{2x}$ and $\sin(2x)$ are linearly independent on the interval $(-\infty, +\infty)$. The example provided uses the Wronskian determinant to show this, but there is some confusion about the calculation. The final calculation shows that the Wronskian of $e^x$ and $\cos(x)$ is $-2e^x\cos(x)$, but there is a request for the Wronskian of $e^{2x}$ and $\cos(2x)$. The conversation then shifts to a different problem, but the final calculation for the Wronskian of $e^x
  • #1
karush
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16.1 Show that $e^{2x}$, sin(2x) are linearly independent on $(-\infty,+\infty)$

https://www.physicsforums.com/attachments/9064
that was the example but...

\begin{align*}
w(e^x,\cos x)&=\left|\begin{array}{rr}e^x&\cos{x} \\ e^x&-\cos{x} \\ \end{array}\right|\\
&=??\\
&=??
\end{align*}
 
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  • #2
$$=e^x(-\cos(x)-\cos(x))=-2e^x\cos(x)\dots$$

But don't you need to calculate $w\left(e^{2x},\cos(2x)\right)?$ What do you get for that?
 
  • #3
karush said:
but...

\begin{align*}
w(e^x,\cos x)&=\left|\begin{array}{rr}e^x&\cos{x} \\ e^x&-\cos{x} \\ \end{array}\right|\\
&=??\\
&=??
\end{align*}

\(\displaystyle w(e^x,\cos{x}) = \begin{vmatrix}
e^x & \cos{x}\\
e^x & -\sin{x}
\end{vmatrix} = -e^x(\sin{x}+\cos{x})\)
 
  • #4
Sorry everybody I think this thread went off the rails
my 2nd post was way off
so Ill post another new one of a similar problem


 
Last edited:

Related to 16.1 Show that e^{2x}, sin(2x) is linearly independent on + infinity -infinity

1. What does it mean for two functions to be linearly independent?

Two functions are said to be linearly independent if neither one can be expressed as a constant multiple of the other. This means that the two functions do not have a linear relationship and cannot be combined to form a third function.

2. How can I show that e^{2x} and sin(2x) are linearly independent?

To show that two functions are linearly independent, you can use the Wronskian determinant. In this case, the Wronskian of e^{2x} and sin(2x) is equal to 2e^{2x}, which is never equal to 0 for any value of x. This means that the two functions are linearly independent.

3. Can two functions be linearly independent on a specific interval but not on another?

Yes, it is possible for two functions to be linearly independent on one interval but not on another. For example, e^{2x} and sin(2x) may be linearly independent on the interval (-infinity, +infinity), but they may not be linearly independent on a smaller interval such as (0, 1).

4. Why is it important to show that two functions are linearly independent?

It is important to show that two functions are linearly independent because it helps us understand their relationship and how they behave. It also allows us to use them as a basis for a larger set of functions, which can be useful in solving differential equations and other mathematical problems.

5. Are there other methods for showing that two functions are linearly independent?

Yes, there are other methods for showing that two functions are linearly independent. One method is to use the definition of linear independence, which states that if the only solution to the equation c1f1(x) + c2f2(x) = 0 is c1 = c2 = 0, then the two functions are linearly independent. Another method is to use the method of reduction of order, which involves substituting one function into the other and solving for the coefficients.

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