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I look for real applications for homogeneous first order ODE's, i.e.

$$ y'(x) = f(x,y) $$ where $$ \exists n\in \mathbb{N} \quad {\rm s.t.} \quad f(\lambda x, \lambda y)= \lambda ^n f(x,y) \quad \forall \lambda \in \mathbb{R} \, \, . $$

This question is very much in the spirit of this post, as I teach this ODE for the second year now, and could find now interesting linkages between it and other fields of math and/or physics. I'm looking for nonlinear cases that cannot be solved via separation of variables, and so the subtitution $v=y/x$ (or its equivalent methods) is neccesary.

Thanks

Community
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Amir Sagiv
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Problem: The easiest one you can imagine. The speed of radium nuclear decay at each subsequent moment of time is proportional to its quantity $ Q $. Find the law of radium decay given that at $ t = t_ {0} $ one had $ Q_ {0} $ amount of radium.

Solution: By the speed of decay we mean the first derivative of the quantity of radium with respect to time, so one has $$\frac{dQ}{dt}=-\lambda{Q}$$ Where the negative sign is taken to account for the fact that the quantity is decreasing. (and this equation is homogeneous as required). The solution is $$Q=ce^{-\lambda{t}}$$ Now we apply the initial conditions to give the constant of integration $$c=Q_{0}e^{\lambda{t_{0}}}$$ So, finally, $$Q(t)=Q_{0}e^{-\lambda(t-t_{0})}$$

Kiryl Pesotski
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  • Thanks! You're right. However, this equation can be solved in an easier manner - it is separable. It does not justify the use of other methods. I edited my OP accrodingly – Amir Sagiv Mar 28 '17 at 12:07