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The Laplace transform is the fundamental operation encoding the canonical ensemble in statistical mechanics. It converts the density of states $d(\varepsilon )$ (a non-statistical concept) into the ca …

You could just modify your second equation once $t>24$, e.g., $$ B'(t) = \theta(24-t) [c (1 - B(t)) \theta(18 - t) - d B(t) S(t) - 3 c (1 - B(t - 18))]-\frac{1}{2}\theta(t-24)B(t) $$ If a smoother mod …

The relationship can be written using the dilation operator as $$ _2 F_1 (a,b,c,p\cdot z) = \exp \left( \ln p \cdot z \frac{\partial}{\partial z} \right)\, {}_2 F_1 (a,b,c,z) $$

The equation has solutions with powers, $f(x) = ax^b$. Inserting this ansatz, one has $$ a b (b-1) x^{b-2} = a (a x^b)^b = a^{b+1} x^{b^2} \ , $$ so the requirements on $a$ and $b$ are $$ b-2 = b^2 \ …

There is no "exact answer" to this question. Answers will contain the words "reasonable" and "appropriate", terms which depend on the context. I'll try to give a reasonable answer. Given a domain of p …

Maybe I'm missing something, but it seems to me you're overthinking this. Since $$ \int_{0}^{1} d\theta (1-\theta ) \theta^{n} = \frac{1}{n+1} -\frac{1}{n+2} = \frac{n!}{(n+2)!} $$ the third term in y …

For the choice $q(x)=2q\cos (2x)$ (with $q$ on the right hand side a constant, I'm trying to stick to standard notation), the differential equation is known as Mathieu's equation, with solutions descr …

One can find a solution of the form $x=y=z$, namely, $x=2$arccot$(\exp (-t-a))$ with the free parameter $a$. Of course, there should be more. Note also the symmetries of the problem: For any solution …