# {y}{\left({x}^{2}-{1}\right)}{\left.{d}{y}\right.}+{x}{\left({y}^{2}+{1}\right)}{\left.{d}{x}\right.}={0}

$${y}{\left({x}^{2}-{1}\right)}{\left.{d}{y}\right.}+{x}{\left({y}^{2}+{1}\right)}{\left.{d}{x}\right.}={0}$$

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Ayesha Gomez

Solve the separable equation x $${\left({y}{\left({x}\right)}^{2}+{1}\right)}+\frac{{{\left.{d}{y}\right.}{\left({x}\right)}}}{{{\left.{d}{x}\right.}}}{\left({x}^{2}-{1}\right)}{y}{\left({x}\right)}={0}:$$
Solve for $$\frac{{{\left.{d}{y}\right.}{\left({x}\right)}}}{{{\left.{d}{x}\right.}}}:$$
$$\frac{{{\left.{d}{y}\right.}{\left({x}\right)}}}{{{\left.{d}{x}\right.}}}=-\frac{{{x}{\left({y}{\left({x}\right)}^{2}+{1}\right)}}}{{{\left({x}^{2}-{1}\right)}{y}{\left({x}\right)}}}$$
Divide both sides by $$-\frac{{{y}{\left({x}\right)}^{2}+{1}}}{{y}}{\left({x}\right)}:$$
$$-\frac{{\frac{{{\left.{d}{y}\right.}{\left({x}\right)}}}{{{\left.{d}{x}\right.}}}{y}{\left({x}\right)}}}{{{y}{\left({x}\right)}^{2}+{1}}}=\frac{x}{{{x}^{2}-{1}}}$$
Integrate both sides with respect to x:
$$\int-\frac{{\frac{{{\left.{d}{y}\right.}{\left({x}\right)}}}{{{\left.{d}{x}\right.}}}{y}{\left({x}\right)}}}{{{y}{\left({x}\right)}^{2}+{1}}}{\left.{d}{x}\right.}=\int\frac{x}{{{x}^{2}-{1}}}{\left.{d}{x}\right.}$$
Evaluate the integrals:
$$-\frac{1}{{2}} \log{{\left({y}{\left({x}\right)}^{2}+{1}\right)}}=\frac{1}{{2}} \log{{\left({x}^{2}-{1}\right)}}+{c}_{{1}}$$, where $$\displaystyle{c}_{{{1}}}$$ is an arbitrary constant.
Solve for y(x):
$${y}{\left({x}\right)}=-\frac{\sqrt{{{e}^{{-{2}{c}_{{1}}}}-{x}^{2}+{1}}}}{\sqrt{{{x}^{2}-{1}}}}{\quad\text{or}\quad}{y}{\left({x}\right)}=\frac{\sqrt{{{e}^{{-{2}{c}_{{1}}}}-{x}^{2}+{1}}}}{\sqrt{{{x}^{2}-{1}}}}$$
INTERMEDIATE STEPS:
Solve for $$y(x)$$:
$$-\frac{1}{{2}} \log{{\left({y}{\left({x}\right)}^{2}+{1}\right)}}=\frac{1}{{2}} \log{{\left({x}^{2}-{1}\right)}}+{c}_{{1}}$$
Hint: Multiply both sides by a constant to simplify the equation.
Multiply both sides by -2:
$$\log{{\left({y}{\left({x}\right)}^{2}+{1}\right)}}=- \log{{\left({x}^{2}-{1}\right)}}-{2}{c}_{{1}}$$
Hint: Eliminate the logarithm from the left hand side.
Cancel logarithms by taking exp of both sides:
$${y}{\left({x}\right)}^{2}+{1}=\frac{{e}^{{-{2}{c}_{{1}}}}}{{{x}^{2}-{1}}}$$
Hint: Isolate terms with y(x) to the left hand side.
Subtract 1 from both sides:
$${y}{\left({x}\right)}^{2}=\frac{{e}^{{-{2}{c}_{{1}}}}}{{{x}^{2}-{1}}}-{1}$$
Hint: Eliminate the exponent on the left hand side.
Take the square root of both sides:
$${y}{\left({x}\right)}=\sqrt{{\frac{{e}^{{-{2}{c}_{{1}}}}}{{{x}^{2}-{1}}}-{1}}}{\quad\text{or}\quad}{y}{\left({x}\right)}=-\sqrt{{\frac{{e}^{{-{2}{c}_{{1}}}}}{{{x}^{2}-{1}}}-{1}}}$$
Simplify the arbitrary constants:
$${y}{\left({x}\right)}=-\frac{\sqrt{{-{x}^{2}+{c}_{{1}}}}}{\sqrt{{{x}^{2}-{1}}}}{\quad\text{or}\quad}{y}{\left({x}\right)}=\frac{\sqrt{{-{x}^{2}+{c}_{{1}}}}}{\sqrt{{{x}^{2}-{1}}}}$$