FP3 Question

Submitted by Anonymous (not verified) on

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$$
\frac{\mathrm{d^{2}y} }{\mathrm{d} x^{2}}+n^2x=0
$$
In the FP3 textbook (AQA) is says the auxiliary equation for this is
$$
k^2+n^2=0
$$
I'm just wondering how you get to this :)

I differentiated with respect

Submitted by O________O (not verified) on

I differentiated with respect to x which leads to $$ k^3+n^2=0 $$
so still a bit lost.

Auxiliary equations

Submitted by Rayman (not verified) on

Substitute $y=e^{kx}$ into the differential equation and you get the auxiliary equation.

You can also do this for the second order differential equations with constant coefficients to get $ak^2 + bk + c = 0$

Actually

Submitted by Rayman (not verified) on

$y= Ae^{kx}$

x and y

Submitted by O________O (not verified) on

would that not only work if the equation was this:
$$ \frac{\mathrm{d^{2}y} }{\mathrm{d} x^{2}}+n^2y=0 $$

My bad.

Submitted by Rayman (not verified) on

I misread the question.
For the other differential equation,
I'm fairly certain you can integrate it twice.
Could it be a typo in the textbook? Should that $x$ have been a $y$?

I thought that too

Submitted by O________O (not verified) on

I thought it was a typo but I found it cropping up somewhere else as well - I doubt it will come up at all but it's a bit weird - Thanks :D

Auxiliary equation

Submitted by stcs on

Certainly a typo. The idea of an auxiliary equation only applies to linear equations with constant coefficients (or related equations). As someone above says, if the equation is really
$$
\frac{d ^2 y}{dx^2} = - n^2 x
$$
you would just integrate twice to get $y= -n^2 x^3/6 + Ax +B
$: no auxiliary equation required.