Calculus Examples Differential Equations

Solve the Differential Equation

$\frac{d y}{d x} = \frac{x^{2}}{1 - y^{2}}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $1 - y^{2}$ .

$(1 - y^{2}) \frac{d y}{d x} = (1 - y^{2}) \frac{x^{2}}{1 - y^{2}}$

Step 1.2

Simplify.

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Step 1.2.1

Simplify the denominator.

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Step 1.2.1.1

Rewrite $1$ as $1^{2}$ .

$(1 - y^{2}) \frac{d y}{d x} = (1 - y^{2}) \frac{x^{2}}{1^{2} - y^{2}}$

Step 1.2.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = y$ .

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

Step 1.2.2

Multiply $1 - y^{2}$ by $\frac{x^{2}}{(1 + y) (1 - y)}$ .

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

Step 1.2.3

Simplify the numerator.

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Step 1.2.3.1

Rewrite $1$ as $1^{2}$ .

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

Step 1.2.3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = y$ .

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

Step 1.2.4

Cancel the common factor of $1 + y$ .

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Step 1.2.4.1

Cancel the common factor.

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

Step 1.2.4.2

Rewrite the expression.

$(1 - y^{2}) \frac{d y}{d x} = \frac{(1 - y) x^{2}}{1 - y}$

Step 1.2.5

Cancel the common factor of $1 - y$ .

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Step 1.2.5.1

Cancel the common factor.

$(1 - y^{2}) \frac{d y}{d x} = \frac{(1 - y) x^{2}}{1 - y}$

Step 1.2.5.2

Divide $x^{2}$ by $1$ .

$(1 - y^{2}) \frac{d y}{d x} = x^{2}$

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int 1 - y^{2} d y = \int x^{2} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the single integral into multiple integrals.

$\int d y + \int - y^{2} d y = \int x^{2} d x$

Step 2.2.2

Apply the constant rule.

$y + C_{1} + \int - y^{2} d y = \int x^{2} d x$

Step 2.2.3

Since $- 1$ is constant with respect to $y$ , move $- 1$ out of the integral.

$y + C_{1} - \int y^{2} d y = \int x^{2} d x$

Step 2.2.4

By the Power Rule, the integral of $y^{2}$ with respect to $y$ is $\frac{1}{3} y^{3}$ .

$y + C_{1} - (\frac{1}{3} y^{3} + C_{2}) = \int x^{2} d x$

Step 2.2.5

Simplify.

$y - \frac{1}{3} y^{3} + C_{3} = \int x^{2} d x$

Step 2.2.6

Reorder terms.

$- \frac{1}{3} y^{3} + y + C_{3} = \int x^{2} d x$

Step 2.3

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$- \frac{1}{3} y^{3} + y + C_{3} = \frac{1}{3} x^{3} + C_{4}$

Step 2.4

Group the constant of integration on the right side as $K$ .

$- \frac{1}{3} y^{3} + y = \frac{1}{3} x^{3} + K$

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