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Consider the system of ODEs:

\begin{align*} \frac{dx}{dt}&=2xy, \\ \frac{dy}{dt}&=x^2+y^2. \end{align*}

I'm struggling to see a sensible way to combine these ODEs in order to solve the system. I've tried, for example, considering dividing one by the other or taking a combination of the two but this doesn't seem to be helping.

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1 Answer 1

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Let $\dot x := \frac{dx}{dt}$ and $\dot y := \frac{dy}{dt}$. From your equations, we have

\begin{align} \dot x + \dot y &= (x + y)^2\\ -(\dot x - \dot y) &= (x - y)^2. \end{align}

Now, using linearity of differentiation, introduce the coordinates $x_+ := x + y$ and $x_- = x - y$, so that the equations become

\begin{align} \dot x_+ &= x_+^2\\ - \dot x_- &= x_-^2. \end{align}

These are easy to solve, and give

\begin{align} x_+ &= \frac{1}{x_{+0} - t_0 - t}\\ x_- &= \frac{1}{t + t_0 - x_{-0}}. \end{align}

Finally, resubstitute $x_+ := x+y$ and $x_- := x-y$ and take combinations of the two to isolate $x$ and $y$.

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