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I'm attempting to solve a problem involving this differential equation:

$$\frac{dy}{dx} = x^2y^2 + x^2 - y^2 - 1$$

Because this is a separable differential equation, I tend to split the equation between variables $x$ and $y$, and integrate both sites; but I can't seem to get them separated, due to the $x^2y^2$.

$$\frac{dy}{dx} + y^2 = x^2y^2 + x^2 - 1$$ $$ dy + y^2 = x^2y^2\space dx + x^2\space dx - dx$$

Now the only $y$ variable left to get on the opposite side is the $y^2$ that's trapped in $x^2y^2$, but extracting $y^2$ would leave a $y^2$ variable in all the other terms. That's sort of the problem.

$$\frac{dy + y^2}{y^2} = x^2 dx + \frac{x^2}{y^2} dx - \frac{dx}{y^2}$$ $$\frac{dy}{y^2} + 1 = x^2 dx + \frac{x^2}{y^2} dx - \frac{dx}{y^2}$$

And now, if I want the $y$ variables out of the left side, I end up to the previous equation.

$$ dy + y^2 = x^2y^2\space dx + x^2\space dx - dx$$

It's a really bad loop I'm in right now. How can I get around this problem, and get each variable to its seperate side for integration?

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2 Answers

up vote 7 down vote accepted

You can factor the right side as $(y^2+1)(x^2-1)$

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Here is a systematic way: assuming your right-hand side, call it $f(x,y) = x^2y^2 + x^2 - y^2 - 1$, actually is separable as $f(x,y) = X(x)Y(y)$, then this will work for any $x_0$ and $y_0$. So arbitrarily choosing $(x_0,y_0) = (0,0)$ for simplicity, we have

$$X(x) \propto \frac{X(x)Y(y_0)}{X(x_0)Y(y_0)} = \frac{f(x,y_0)}{f(x_0,y_0)} = \frac{x^2 - 1}{-1} = 1-x^2.$$

Now with* $X(x)=1-x^2$, we get the other factor

$$Y(y) \propto X(x_0)Y(y) = f(x_0,y) = -y^2 - 1$$

and lo and behold, $X(x) = 1-x^2$ and* $Y(y) = -y^2 - 1$ give $f(x,y)$.

*In taking $X(x) = 1-x^2$, I threw out the constant of proportionality $1/X(x_0)$, and in taking $Y(y) = -y^2 - 1$, I threw out $X(x_0)$. But it's OK since they would have cancelled out anyway. I discovered this shortcut after getting tired of guessing on this type of problem.

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