Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Solve for $\frac{d y}{d x}$ .

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

Apply the distributive property.

$x^{2} \frac{d y}{d x} + 1 \frac{d y}{d x} + y^{2} + 1 = 0$

Step 1.1.1.2

Multiply $\frac{d y}{d x}$ by $1$ .

$x^{2} \frac{d y}{d x} + \frac{d y}{d x} + y^{2} + 1 = 0$

Step 1.1.2

Move all terms not containing $\frac{d y}{d x}$ to the right side of the equation.

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Step 1.1.2.1

Subtract $y^{2}$ from both sides of the equation.

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

Step 1.1.2.2

Subtract $1$ from both sides of the equation.

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

Step 1.1.3

Factor $\frac{d y}{d x}$ out of $x^{2} \frac{d y}{d x} + \frac{d y}{d x}$ .

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Step 1.1.3.1

Factor $\frac{d y}{d x}$ out of $x^{2} \frac{d y}{d x}$ .

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

Step 1.1.3.2

Raise $\frac{d y}{d x}$ to the power of $1$ .

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

Step 1.1.3.3

Factor $\frac{d y}{d x}$ out of ${(\frac{d y}{d x})}^{1}$ .

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

Step 1.1.3.4

Factor $\frac{d y}{d x}$ out of $\frac{d y}{d x} x^{2} + \frac{d y}{d x} \cdot 1$ .

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

Step 1.1.4

Divide each term in $\frac{d y}{d x} (x^{2} + 1) = - y^{2} - 1$ by $x^{2} + 1$ and simplify.

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Step 1.1.4.1

Divide each term in $\frac{d y}{d x} (x^{2} + 1) = - y^{2} - 1$ by $x^{2} + 1$ .

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

Step 1.1.4.2

Simplify the left side.

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Step 1.1.4.2.1

Cancel the common factor of $x^{2} + 1$ .

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Step 1.1.4.2.1.1

Cancel the common factor.

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

Step 1.1.4.2.1.2

Divide $\frac{d y}{d x}$ by $1$ .

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

Step 1.1.4.3

Simplify the right side.

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Step 1.1.4.3.1

Combine the numerators over the common denominator.

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

Step 1.1.4.3.2

Factor $- 1$ out of $- y^{2}$ .

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

Step 1.1.4.3.3

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 1.1.4.3.4

Factor $- 1$ out of $- (y^{2}) - 1 (1)$ .

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

Step 1.1.4.3.5

Simplify the expression.

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Step 1.1.4.3.5.1

Rewrite $- (y^{2} + 1)$ as $- 1 (y^{2} + 1)$ .

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

Step 1.1.4.3.5.2

Move the negative in front of the fraction.

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

Step 1.2

Multiply both sides by $\frac{1}{y^{2} + 1}$ .

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

Step 1.3

Simplify.

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Step 1.3.1

Rewrite using the commutative property of multiplication.

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

Step 1.3.2

Cancel the common factor of $y^{2} + 1$ .

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Step 1.3.2.1

Move the leading negative in $- \frac{1}{y^{2} + 1}$ into the numerator.

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

Step 1.3.2.2

Cancel the common factor.

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

Step 1.3.2.3

Rewrite the expression.

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

Step 1.4

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Simplify the expression.

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Step 2.2.1.1

Reorder $y^{2}$ and $1$ .

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

Step 2.2.1.2

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

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

Step 2.2.2

The integral of $\frac{1}{1^{2} + y^{2}}$ with respect to $y$ is $\arctan (y) + C_{1}$ .

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

Step 2.3

Integrate the right side.

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Step 2.3.1

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

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

Step 2.3.2

Simplify the expression.

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Step 2.3.2.1

Reorder $x^{2}$ and $1$ .

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

Step 2.3.2.2

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

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

Step 2.3.3

The integral of $\frac{1}{1^{2} + x^{2}}$ with respect to $x$ is $\arctan (x) + C_{2}$ .

$\arctan (y) + C_{1} = - (\arctan (x) + C_{2})$

Step 2.3.4

Simplify.

$\arctan (y) + C_{1} = - \arctan (x) + C_{2}$

Step 2.4

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

$\arctan (y) = - \arctan (x) + K$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $- \arctan (x) + K = \arctan (y)$ .

$- \arctan (x) + K = \arctan (y)$

Step 3.2

Subtract $K$ from both sides of the equation.

$- \arctan (x) = \arctan (y) - K$

Step 3.3

Divide each term in $- \arctan (x) = \arctan (y) - K$ by $- 1$ and simplify.

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Step 3.3.1

Divide each term in $- \arctan (x) = \arctan (y) - K$ by $- 1$ .

$\frac{- \arctan (x)}{- 1} = \frac{\arctan (y)}{- 1} + \frac{- K}{- 1}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Dividing two negative values results in a positive value.

$\frac{\arctan (x)}{1} = \frac{\arctan (y)}{- 1} + \frac{- K}{- 1}$

Step 3.3.2.2

Divide $\arctan (x)$ by $1$ .

$\arctan (x) = \frac{\arctan (y)}{- 1} + \frac{- K}{- 1}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Simplify each term.

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Step 3.3.3.1.1

Move the negative one from the denominator of $\frac{\arctan (y)}{- 1}$ .

$\arctan (x) = - 1 \cdot \arctan (y) + \frac{- K}{- 1}$

Step 3.3.3.1.2

Rewrite $- 1 \cdot \arctan (y)$ as $- \arctan (y)$ .

$\arctan (x) = - \arctan (y) + \frac{- K}{- 1}$

Step 3.3.3.1.3

Dividing two negative values results in a positive value.

$\arctan (x) = - \arctan (y) + \frac{K}{1}$

Step 3.3.3.1.4

Divide $K$ by $1$ .

$\arctan (x) = - \arctan (y) + K$

Step 3.4

Take the inverse arctangent of both sides of the equation to extract $y$ from inside the arctangent.

$x = \tan (- \arctan (y) + K)$

Step 3.5

Rewrite the equation as $\tan (- \arctan (y) + K) = x$ .

$\tan (- \arctan (y) + K) = x$

Step 3.6

Take the inverse tangent of both sides of the equation to extract $\arctan (y)$ from inside the tangent.

$- \arctan (y) + K = \arctan (x)$

Step 3.7

Subtract $K$ from both sides of the equation.

$- \arctan (y) = \arctan (x) - K$

Step 3.8

Divide each term in $- \arctan (y) = \arctan (x) - K$ by $- 1$ and simplify.

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Step 3.8.1

Divide each term in $- \arctan (y) = \arctan (x) - K$ by $- 1$ .

$\frac{- \arctan (y)}{- 1} = \frac{\arctan (x)}{- 1} + \frac{- K}{- 1}$

Step 3.8.2

Simplify the left side.

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Step 3.8.2.1

Dividing two negative values results in a positive value.

$\frac{\arctan (y)}{1} = \frac{\arctan (x)}{- 1} + \frac{- K}{- 1}$

Step 3.8.2.2

Divide $\arctan (y)$ by $1$ .

$\arctan (y) = \frac{\arctan (x)}{- 1} + \frac{- K}{- 1}$

Step 3.8.3

Simplify the right side.

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Step 3.8.3.1

Simplify each term.

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Step 3.8.3.1.1

Move the negative one from the denominator of $\frac{\arctan (x)}{- 1}$ .

$\arctan (y) = - 1 \cdot \arctan (x) + \frac{- K}{- 1}$

Step 3.8.3.1.2

Rewrite $- 1 \cdot \arctan (x)$ as $- \arctan (x)$ .

$\arctan (y) = - \arctan (x) + \frac{- K}{- 1}$

Step 3.8.3.1.3

Dividing two negative values results in a positive value.

$\arctan (y) = - \arctan (x) + \frac{K}{1}$

Step 3.8.3.1.4

Divide $K$ by $1$ .

$\arctan (y) = - \arctan (x) + K$

Step 3.9

Take the inverse arctangent of both sides of the equation to extract $y$ from inside the arctangent.

$y = \tan (- \arctan (x) + K)$

Step 3.10

Since $y$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\tan (- \arctan (y) + K) = x$

Step 3.11

Take the inverse tangent of both sides of the equation to extract $\arctan (y)$ from inside the tangent.

$- \arctan (y) + K = \arctan (x)$

Step 3.12

Subtract $K$ from both sides of the equation.

$- \arctan (y) = \arctan (x) - K$

Step 3.13

Divide each term in $- \arctan (y) = \arctan (x) - K$ by $- 1$ and simplify.

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Step 3.13.1

Divide each term in $- \arctan (y) = \arctan (x) - K$ by $- 1$ .

$\frac{- \arctan (y)}{- 1} = \frac{\arctan (x)}{- 1} + \frac{- K}{- 1}$

Step 3.13.2

Simplify the left side.

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Step 3.13.2.1

Dividing two negative values results in a positive value.

$\frac{\arctan (y)}{1} = \frac{\arctan (x)}{- 1} + \frac{- K}{- 1}$

Step 3.13.2.2

Divide $\arctan (y)$ by $1$ .

$\arctan (y) = \frac{\arctan (x)}{- 1} + \frac{- K}{- 1}$

Step 3.13.3

Simplify the right side.

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Step 3.13.3.1

Simplify each term.

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Step 3.13.3.1.1

Move the negative one from the denominator of $\frac{\arctan (x)}{- 1}$ .

$\arctan (y) = - 1 \cdot \arctan (x) + \frac{- K}{- 1}$

Step 3.13.3.1.2

Rewrite $- 1 \cdot \arctan (x)$ as $- \arctan (x)$ .

$\arctan (y) = - \arctan (x) + \frac{- K}{- 1}$

Step 3.13.3.1.3

Dividing two negative values results in a positive value.

$\arctan (y) = - \arctan (x) + \frac{K}{1}$

Step 3.13.3.1.4

Divide $K$ by $1$ .

$\arctan (y) = - \arctan (x) + K$

Step 3.14

Take the inverse arctangent of both sides of the equation to extract $y$ from inside the arctangent.

$y = \tan (- \arctan (x) + K)$