# Math Help - why? solution to second order DE (d2y/dx2)-a2y=0 is in form y=Asinh(ax)+Bcosh(ax)

1. ## why? solution to second order DE (d2y/dx2)-a2y=0 is in form y=Asinh(ax)+Bcosh(ax)

How do i show this mathematically, preferably without using auxillary equations, that the solution to second order DE (d2y/dx2)-a2y=0 is in form y=Asinh(ax)+Bcosh(ax)

2. ## Re: why? solution to second order DE (d2y/dx2)-a2y=0 is in form y=Asinh(ax)+Bcosh(ax)

I think you can use the substitution: $y=e^{tx}$.

3. ## Re: why? solution to second order DE (d2y/dx2)-a2y=0 is in form y=Asinh(ax)+Bcosh(ax)

where to from there? I end up with:

$y\left(\left(\frac{\ln{y}}{x}\right)^2-9\right)=0\\\\y=e^{\pm3x}$

I'm not sure if this is right. There are no general constants !!!

4. ## Re: why? solution to second order DE (d2y/dx2)-a2y=0 is in form y=Asinh(ax)+Bcosh(ax)

I would solve it like this:
Let $y=e^{tx}$
Substituting in the equation gives:
$t^2\cdot e^{tx}-a^2\cdot e^{tx}=0$
$t^2-a^2=0 \Leftrightarrow (t-a)(t+a)=0$
$t=a \ \mbox{or} \ t=-a$
So the general solution is:
$y=c_1\cdot e^{ax}+c_2\cdot e^{-ax}$
Where $c_1,c_2$ are constants (which can be chosen).

5. ## Re: why? solution to second order DE (d2y/dx2)-a2y=0 is in form y=Asinh(ax)+Bcosh(ax)

Originally Posted by Siron
I would solve it like this:
Let $y=e^{tx}$
Substituting in the equation gives:
$t^2\cdot e^{tx}-a^2\cdot e^{tx}=0$
$t^2-a^2=0 \Leftrightarrow (t-a)(t+a)=0$
$t=a \ \mbox{or} \ t=-a$
So the general solution is:
$y=c_1\cdot e^{ax}+c_2\cdot e^{-ax}$
Where $c_1,c_2$ are constants (which can be chosen).
This is using the fact that any de with n solutions will also have solutions that are a linear combination of one or more of those n solutions, right? Would anyone know where there is a proof for this fact?

6. ## Re: why? solution to second order DE (d2y/dx2)-a2y=0 is in form y=Asinh(ax)+Bcosh(ax)

Originally Posted by Istafa
How do i show this mathematically, preferably without using auxillary equations
One of the most 'attracting' properties of the hyperboloc function is ...

$\frac{d}{dx} \sinh a x = a\ \cosh a x$ (1)

$\frac{d}{dx} \cosh a x = a\ \sinh a x$ (2)

... and from (1) and (2)...

$\frac{d^{2}}{d x^{2}} \sinh a x = a^{2}\ \sinh a x$ (3)

$\frac{d^{2}}{d x^{2}} \cosh a x = a^{2}\ \cosh a x$ (4)

... so that both $\sinh a x$ and $\cosh a x$ satisfy the ODE...

$\frac{d^{2} y}{d x^{2}} = a^{2}\ y$ (5)

But the (5) is a linear second order ODE and any linear combination of two linearly independent solutions is also a solution...

Kind regards

$\chi$ $\sigma$

7. ## Re: why? solution to second order DE (d2y/dx2)-a2y=0 is in form y=Asinh(ax)+Bcosh(ax)

If the problem was, as initially stated, NOT to solve $y''- a^2y= 0$ but simply to show that y(x)= Asinh(ax)+ Bcosh(ax) is a solution, then you simply need to note that $(Acosh(ax)+ Bsinh(ax))''= Aa^2cosh(ax)+ Ba^2sinh(ax)$, as chisigma said, and that is $a^2(Acosh(ax)+ Bsinh(ax))= y$ so that $y''- a^2y= a^2y- a^2y= 0$.

By the way, just as sin(ax) and cos(ax) are solutions to the initial value problems $y''+ a^2y= 0$, y(0)= 0, y'(0)= 1 and $y''+ a^2y= 0$, y(0)= 1, y'(0)= 0, respectively, so sinh(ax) and cosh(ax) are solutions to the initial value problems $y''- a^2y= 0$, y(0)= 0, y'(0)= 1 and $y''- a^2y= 0$, y(0)= 1, y'(0)= 0, respectively.