How Do You Solve for Alpha and Beta in Exponential Equations?

[Dustinsfl]

\begin{align}
\alpha_nb^n +\beta_nb^{-n}=A_n\\
\alpha_na^n +\beta_na^{-n}=C_n
\end{align}

How does one go from that to
$$
\alpha_n = \frac{A_n/a_n - C_n/b^n}{(b/a)^n-(a/b)^n}
$$
and
$$
\beta_n = \frac{a^nC_n - b^nA_n}{(b/a)^n-(a/b)^n}
$$

 

[chisigma]

\begin{align}
\alpha_nb^n +\beta_nb^{-n}=A_n\\
\alpha_na^n +\beta_na^{-n}=C_n
\end{align}

How does one go from that to
$$
\alpha_n = \frac{A_n/a_n - C_n/b^n}{(b/a)^n-(a/b)^n}
$$
and
$$
\beta_n = \frac{a^nC_n - b^nA_n}{(b/a)^n-(a/b)^n}
$$

That's a [linear...] system of two equation in the unknown variables $\alpha_{n}$ and $\beta_{n}$...

Kind regards

$\chi$ $\sigma$

 

[Dustinsfl]

That's a [linear...] system of two equation in the unknown variables $\alpha_{n}$ and $\beta_{n}$...

Kind regards

$\chi$ $\sigma$

I do know that, but every time I solve it, I don't get the correct answer. I even put it into full version Mathematica and Wolfram online and it just returns an error.

For $\beta$, I keep getting
$$
\beta_n = \frac{b^nC_n - a^nA_n}{\text{the correct denominator}}
$$

 

[MarkFL]

I would use elimination, for example, multiplying the second equation by:

$\displaystyle -\left(\frac{a}{b} \right)^n$

gives us:

$\displaystyle -\alpha_na^{2n}b^{-n}-\beta_nb^{-n}=-C_na^nb^{-n}$

Adding this to the first equation, we find:

$\displaystyle \alpha_n(b^n-a^{2n}b^{-n})=A_n-C_na^nb^{-n}$

$\displaystyle \alpha_n=\frac{A_n-C_na^nb^{-n}}{b^n-a^{2n}b^{-n}}=\frac{\frac{A_n}{a^n}-\frac{C_n}{b^{n}}}{\left(\frac{b}{a} \right)^n-\left(\frac{a}{b} \right)^n}$

The solution for $\displaystyle \beta_n$ can now be found by substitution.

 

[CellCoree]

To solve for alpha and beta in this system of equations, we can use the method of elimination. We can eliminate the variable beta by multiplying the first equation by b^n and the second equation by a^n, resulting in:

\begin{align}
\alpha_nb^{2n}+\beta_n&=A_nb^n\\
\alpha_na^{2n}+\beta_n&=C_na^n
\end{align}

Subtracting these equations, we can eliminate beta and solve for alpha:

\begin{align}
\alpha_na^{2n}-\alpha_nb^{2n}&=C_na^n-A_nb^n\\
\alpha_n(a^{2n}-b^{2n})&=C_na^n-A_nb^n\\
\alpha_n&=\frac{C_na^n-A_nb^n}{a^{2n}-b^{2n}}
\end{align}

We can then substitute this value back into one of the original equations to solve for beta. Let's substitute it into the first equation:

\begin{align}
\frac{C_na^n-A_nb^n}{a^{2n}-b^{2n}}b^n +\beta_nb^{-n}&=A_n\\
\frac{C_na^n-A_nb^n}{a^{n}-b^{n}} +\beta_nb^{-n}&=A_n\\
\beta_nb^{-n}&=A_n-\frac{C_na^n-A_nb^n}{a^{n}-b^{n}}\\
\beta_n&=\frac{a^nA_n-a^{2n}C_n+a^{2n}A_n-b^{2n}A_n}{a^{2n}-b^{2n}}\\
\beta_n&=\frac{a^nC_n-a^{2n}C_n+a^{2n}A_n-b^{2n}A_n}{a^{2n}-b^{2n}}\\
\beta_n&=\frac{a^nC_n-b^nA_n}{a^{2n}-b^{2n}}\\
\end{align}

This can be simplified further by factoring out a^n from the numerator and denominator:

\begin{align}
\beta_n&=\frac{a^n(C_n-b^nA_n)}{a^n(a^n-b^n)}\\
\beta_n&=\frac{C_n-b^nA_n}{a^n-b^n}
\