Calculus Examples

$x^{2} y \frac{d y}{d x} = e^{y}$

Step 1

Separate the variables.

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Step 1.1

Divide each term in $x^{2} y \frac{d y}{d x} = e^{y}$ by $x^{2} y$ and simplify.

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Step 1.1.1

Divide each term in $x^{2} y \frac{d y}{d x} = e^{y}$ by $x^{2} y$ .

$\frac{x^{2} y \frac{d y}{d x}}{x^{2} y} = \frac{e^{y}}{x^{2} y}$

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

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

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Step 1.1.2.1.1

Cancel the common factor.

$\frac{x^{2} y \frac{d y}{d x}}{x^{2} y} = \frac{e^{y}}{x^{2} y}$

Step 1.1.2.1.2

Rewrite the expression.

$\frac{y \frac{d y}{d x}}{y} = \frac{e^{y}}{x^{2} y}$

Step 1.1.2.2

Cancel the common factor of $y$ .

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Step 1.1.2.2.1

Cancel the common factor.

$\frac{y \frac{d y}{d x}}{y} = \frac{e^{y}}{x^{2} y}$

Step 1.1.2.2.2

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

$\frac{d y}{d x} = \frac{e^{y}}{x^{2} y}$

Step 1.2

Regroup factors.

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

Step 1.3

Multiply both sides by $\frac{y}{e^{y}}$ .

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

Step 1.4

Simplify.

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Step 1.4.1

Combine.

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

Step 1.4.2

Combine.

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

Step 1.4.3

Cancel the common factor of $y$ .

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Step 1.4.3.1

Cancel the common factor.

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

Step 1.4.3.2

Rewrite the expression.

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

Step 1.4.4

Cancel the common factor of $e^{y}$ .

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Step 1.4.4.1

Cancel the common factor.

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

Step 1.4.4.2

Rewrite the expression.

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

Step 1.5

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

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

Negate the exponent of $e^{y}$ and move it out of the denominator.

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

Step 2.2.1.2

Multiply the exponents in ${(e^{y})}^{- 1}$ .

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Step 2.2.1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.2.1.2.2

Move $- 1$ to the left of $y$ .

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

Step 2.2.1.2.3

Rewrite $- 1 y$ as $- y$ .

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

Step 2.2.2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = y$ and $d v = e^{- y}$ .

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

Step 2.2.3

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

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

Step 2.2.4

Simplify.

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Step 2.2.4.1

Multiply $- 1$ by $- 1$ .

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

Step 2.2.4.2

Multiply $\int e^{- y} d y$ by $1$ .

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

Step 2.2.5

Let $u = - y$ . Then $d u = - d y$ , so $- d u = d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.5.1

Let $u = - y$ . Find $\frac{d u}{d y}$ .

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Step 2.2.5.1.1

Differentiate $- y$ .

$\frac{d}{d y} [- y]$

Step 2.2.5.1.2

Since $- 1$ is constant with respect to $y$ , the derivative of $- y$ with respect to $y$ is $- \frac{d}{d y} [y]$ .

$- \frac{d}{d y} [y]$

Step 2.2.5.1.3

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 1$ .

$- 1 \cdot 1$

Step 2.2.5.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 2.2.5.2

Rewrite the problem using $u$ and $d u$ .

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

Step 2.2.6

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

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

Step 2.2.7

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

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

Step 2.2.8

Rewrite $y (- e^{- y}) - (e^{u} + C_{1})$ as $- y e^{- y} - e^{u} + C_{1}$ .

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

Step 2.2.9

Replace all occurrences of $u$ with $- y$ .

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

Step 2.2.10

Reorder terms.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Apply basic rules of exponents.

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Step 2.3.1.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

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

Step 2.3.1.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 2.3.1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.3.1.2.2

Multiply $2$ by $- 1$ .

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

Step 2.3.2

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

$- e^{- y} y - e^{- y} + C_{1} = - x^{- 1} + C_{2}$

Step 2.3.3

Rewrite $- x^{- 1} + C_{2}$ as $- \frac{1}{x} + C_{2}$ .

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

Step 2.4

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

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