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I have eqn: $\frac{dx}{dt} = -y(t)$ and $\frac{dy}{dt} = x(t)$

I know that $(x(0),y(0))= (1,0)$.

I want to solve eqn and show that it admits an invariant $I = x(t)^2 + y(t)^2$.

I know $x' = -y$,

$y' = x$,

$x^{\prime\prime} = -y' = -x$

I know general solution of $x" = -x$ is $x = a\sin x = b\cos x$.

I know $x(0) = a\sin 0 + b\cos 0 = 1$ So $b = 1$

How can I show $a = 1$? (I think it should!)

I tried $x' = a\cos x - b\sin x$ since $y = -x$ but it just gives $ a = 0$.

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    Please, try to make the title of your question more informative. E.g., *Why does a imply a+c?* is much more useful for other users than *A question about inequality.* From [How can I ask a good question?](http://meta.math.stackexchange.com/a/589/): *Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader.*2012-11-18

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You have $\frac{dx}{dt} = -y(t)$ and $x(t)=a\sin(t)+\cos(t)$ so take the derivative and use $y(0)=0$.

You will find $a=0$, as you have already discovered but do not believe. If you had $a=1$ then you would not have $y(0)=0$.

So you have $(x(t),y(t)) = \left((\cos(t),-\sin(t)\right)$. This is a parametric equation of a circle of radius $1$ centred on the origin.

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    @sam (a) It is not true unless it involves squares (Pythagoras). (b) Your question said "it admits an invariant $I = x(t)^2 + y(t)^2$."2012-11-19