Find out the answer to "y(x2−1)dy+x(y2+1)dx=0"

Tazmin Horton

Answered question

2021-09-09

$y (x^{2} - 1) d y + x (y^{2} + 1) d x = 0$

Answer & Explanation

Ayesha Gomez

Skilled2021-09-10Added 104 answers

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

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