Solving Harmonic Oscillation w/ BC y(1)=B

[knockout_artist]

Homework Statement

[/B]
For differential equation of the form
$y''- y = 0$
BC is
$y(1) = B$

which usually have general solution

$y(x) = C1 e^x + C2 e^{-x}$

But this manual I am reading always want to go with general solution

$y = C1 \cosh(x) + C2 \sinh( x)$

I assume that is also acceptable solution.

Homework Equations

When in this manual boundary condition applied on cosh and sinh in the general solution equation.
It looks like this
$y(1) = B cosh(1-x) + sinh(1- x)$
BC was y(1) = B3. The Attempt at a Solution

I would think that it should be
$y(1) = B cosh(1) + sinh(1)$

Why they do (x-1) ?

 

[kuruman]

I assume that is also acceptable solution.

It is the same solution with C₁ and C₂ redefined. Can you see how?

y(1) = B cosh(1-x) + sinh(1- x)

I can't parse what you mean by this. The term on the left, $y(1)$, must be a constant not a function of $x$.

 

[Ray Vickson]

Homework Statement

[/B]
For differential equation of the form
$y''- y = 0 ; y(1) = B <-- BC$

which usually have general solution

$y(x) = C1 e^x + C2 e^{-x}$

But this manual I am reading always want to go with general solution

$y = C1 cosh(x) + C2 sinh( x)$

I assume that is also acceptable solution.

Homework Equations

When in this manual boundary condition applied on cosh and sinh in the general solution equation.
It looks like this
$y(1) = B cosh(1-x) + sinh(1- x)$
BC was y(1) = B3. The Attempt at a Solution

I would think that it should be
$y(1) = B cosh(1) + sinh(1)$

Why they do (x-1) ?

Please! Stop writing $cosh(x)$ and $sinh(x)$; instead, write $\cosh(x)$ and $\sinh(x)$. You do that by typing "\cosh" and "\sinh" instead of "cosh" and "sinh". $cosh(x)$ looks ugly and is hard to read, but $\cosh(x)$ (or even $\cosh x$) looks good. That is the way LaTeX was designed to work,

Also: what did the notation $y(1) = B <-- BC$ mean? I get the $y(1) = B$ part, but what is $<--BC?$. And, just so you know: the way to make a backwards arrow in LaTeX is to write $\leftarrow$ (" \_nospace_leftarrow").