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Jacob R.
Chemistry 101
1 week, 3 days ago
Solve $\frac{d^{2} \theta}{d t^{2}}+a^{2} \theta=0$ subject to the condition $\theta(t+2 \pi \tau)=\theta(t)$.
Chapter 12
Second-order differential equations. Constant coefficients
in this question we've been given a second order differential equation where we've got the second deliver t of plus a squared multiplied by Data being equal to zero. And we've been told that we've got the value of data. T Plus two Pi child being equal to data as a function of team. Now looking at this, the general solution to this differential equation is theater as a function of team is equality A multiplied by The exponent of I-80 plus B. Multiplied by the exponent of negative I N. Team. Now looking at this who have been given these cyclic boundaries and if we actually apply them to this general solution to determine the constants A and B, what we are going to get is we are saying peter T plus two pi tau is equal to a. The exponent of I A T Plus two Pi. How plus B. The exponent of negative I A team last two A top. So what this means is we've got data. This right here has been equal to detail or rather a the exponent of I 80 multiplied by the exponent of I two pi A tao plus B. The exponent of negative I a t plus two. Mm hmm. Top. So if we are to simplify this would help this being equal to detail. Town. The top is a function of T. When we've got the exponent of Plus or -1 two pi a child being equal Toral. What now that we have this we are saying this condition is satisfied when we've got two pi and tau being equal to two pi N for any integer value of. And so we've got a being equal to and divided by. So what this means is the solution. We've got better as a function of T being equal to a i n t over town plus B e i m t over how where we've got em Being equal to zero plus or -1 Plus or -2. It is.
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Chemistry is the science of matter, especially its chemical reactions, but also its composition, structure and properties. Chemistry deals with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds. Chemistry also involves understanding the properties and interactions of individual atoms and molecules for use in larger-scale applications.
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