# Solve: xdy/dx=1-y^2

Question
Differential equations
Solve: $$\displaystyle{x}\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}={1}-{y}^{{2}}$$

2021-01-23
$$\displaystyle{x}\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}={1}-{y}^{{2}}$$
cross-multiply to get,
$$\displaystyle\frac{{1}}{{{1}-{y}^{{2}}}}{\left.{d}{y}\right.}=\frac{{1}}{{x}}{\left.{d}{x}\right.}$$
integrate both sides,
$$\displaystyle\int\frac{{1}}{{{1}-{y}^{{2}}}}{\left.{d}{y}\right.}=\int\frac{{1}}{{x}}{\left.{d}{x}\right.}$$
$$\displaystyle{a}{\text{tanh}{{\left({y}\right)}}}={\ln{{\left({A}{x}\right)}}}$$
$$\displaystyle{y}={\text{tanh}{{\left({\ln{{\left({A}{x}\right)}}}\right)}}}$$

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