Suppose we have some ODE given by [itex]y' = G(x,y)/H(x,y)[/itex]. Let x and y depend on a third variable, t, so that x and y are parametrized in a way. Then applying the chain rule to y' gives(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{G(x,y)}{H(x,y)}[/tex]

Then comparing the numerators and denominators gives a system of equations for x and y: [itex]dy/dt = G(x,y); dx/dt = H(x,y).[/itex] Then you can solve this system of equations for x and y to get a very, very implicit solution to the problem.

An example for this method is the following differential equation:

[tex]y' = \frac{x+3y-5}{x-y-1}[/tex]

If the constants weren't there, this equation would be homogeneous. However, with them, the method I described above can be implemented. One would then need to solve the system

[tex]\vec{y}' = \left( \begin{array}{cc} 1 & 3 \\ 1 & -1 \end{array} \right)]\vec{y} - \left( \begin{array}{c}5 \\ 1 \end{array} \right)[/tex]

where [itex]\vec{y} = (y, x)^T[/itex]. Solving this system gives [itex]y(t) = 3c_1e^{2t} + c_2e^{-2t} - 5[/itex] and [itex]x(t) = c_1e^{2t} - c_2e^{-2t} - 1.[/itex]

This doesn't give y as a function of x unfortunately, but it does give a solution, however implicit. Is this legitimate, or is there a better method?

edit: Incidentally, you can actually get this particular solution in terms of x. Note that [itex]x(t) = 2\sinh(2t) - 1 \Rightarrow \frac{x+1}{2} = \sinh(2t) \Rightarrow t = \frac{1}{2}\sinh^{-1}\left(\frac{x+1}{2}\right)[/itex]. You can plug this into the y equation to get a horrific expression for y in terms of x, but this won't be the case in general.

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# Is this a legitimate method for solving first order ODEs in fractional form?

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