Calculus Examples

$\frac{d y}{d x} = (x - 3) e^{- 2 y}$

Step 1

Separate the variables.

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Step 1.1

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

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

Step 1.2

Cancel the common factor of $e^{- 2 y}$ .

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Step 1.2.1

Factor $e^{- 2 y}$ out of $(x - 3) e^{- 2 y}$ .

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

Step 1.2.2

Cancel the common factor.

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

Step 1.2.3

Rewrite the expression.

$\frac{1}{e^{- 2 y}} \frac{d y}{d x} = x - 3$

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

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 $e^{- 2 y}$ and move it out of the denominator.

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

Step 2.2.1.2

Simplify.

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Step 2.2.1.2.1

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

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Step 2.2.1.2.1.1

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

$\int 1 e^{- 2 y \cdot - 1} d y = \int x - 3 d x$

Step 2.2.1.2.1.2

Multiply $- 1$ by $- 2$ .

$\int 1 e^{2 y} d y = \int x - 3 d x$

Step 2.2.1.2.2

Multiply $e^{2 y}$ by $1$ .

$\int e^{2 y} d y = \int x - 3 d x$

Step 2.2.2

Let $u = 2 y$ . Then $d u = 2 d y$ , so $\frac{1}{2} d u = d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.2.1

Let $u = 2 y$ . Find $\frac{d u}{d y}$ .

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Step 2.2.2.1.1

Differentiate $2 y$ .

$\frac{d}{d y} [2 y]$

Step 2.2.2.1.2

Since $2$ is constant with respect to $y$ , the derivative of $2 y$ with respect to $y$ is $2 \frac{d}{d y} [y]$ .

$2 \frac{d}{d y} [y]$

Step 2.2.2.1.3

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

$2 \cdot 1$

Step 2.2.2.1.4

Multiply $2$ by $1$ .

$2$

Step 2.2.2.2

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

$\int e^{u} \frac{1}{2} d u = \int x - 3 d x$

Step 2.2.3

Combine $e^{u}$ and $\frac{1}{2}$ .

$\int \frac{e^{u}}{2} d u = \int x - 3 d x$

Step 2.2.4

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int e^{u} d u = \int x - 3 d x$

Step 2.2.5

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

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

Step 2.2.6

Simplify.

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

Step 2.2.7

Replace all occurrences of $u$ with $2 y$ .

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

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

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

Step 2.3.3

Apply the constant rule.

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

Step 2.3.4

Simplify.

$\frac{1}{2} e^{2 y} + C_{1} = \frac{1}{2} x^{2} - 3 x + C_{4}$

Step 2.4

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

$\frac{1}{2} e^{2 y} = \frac{1}{2} x^{2} - 3 x + K$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} e^{2 y}) = 2 (\frac{1}{2} x^{2} - 3 x + K)$

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $2 (\frac{1}{2} e^{2 y})$ .

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Step 3.2.1.1.1

Combine $\frac{1}{2}$ and $e^{2 y}$ .

$2 \frac{e^{2 y}}{2} = 2 (\frac{1}{2} x^{2} - 3 x + K)$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

$2 \frac{e^{2 y}}{2} = 2 (\frac{1}{2} x^{2} - 3 x + K)$

Step 3.2.1.1.2.2

Rewrite the expression.

$e^{2 y} = 2 (\frac{1}{2} x^{2} - 3 x + K)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $2 (\frac{1}{2} x^{2} - 3 x + K)$ .

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Step 3.2.2.1.1

Combine $\frac{1}{2}$ and $x^{2}$ .

$e^{2 y} = 2 (\frac{x^{2}}{2} - 3 x + K)$

Step 3.2.2.1.2

Apply the distributive property.

$e^{2 y} = 2 \frac{x^{2}}{2} + 2 (- 3 x) + 2 K$

Step 3.2.2.1.3

Simplify.

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Step 3.2.2.1.3.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.3.1.1

Cancel the common factor.

$e^{2 y} = 2 \frac{x^{2}}{2} + 2 (- 3 x) + 2 K$

Step 3.2.2.1.3.1.2

Rewrite the expression.

$e^{2 y} = x^{2} + 2 (- 3 x) + 2 K$

Step 3.2.2.1.3.2

Multiply $- 3$ by $2$ .

$e^{2 y} = x^{2} - 6 x + 2 K$

Step 3.3

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{2 y}) = \ln (x^{2} - 6 x + 2 K)$

Step 3.4

Expand the left side.

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Step 3.4.1

Expand $\ln (e^{2 y})$ by moving $2 y$ outside the logarithm.

$2 y \ln (e) = \ln (x^{2} - 6 x + 2 K)$

Step 3.4.2

The natural logarithm of $e$ is $1$ .

$2 y \cdot 1 = \ln (x^{2} - 6 x + 2 K)$

Step 3.4.3

Multiply $2$ by $1$ .

$2 y = \ln (x^{2} - 6 x + 2 K)$

Step 3.5

Divide each term in $2 y = \ln (x^{2} - 6 x + 2 K)$ by $2$ and simplify.

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Step 3.5.1

Divide each term in $2 y = \ln (x^{2} - 6 x + 2 K)$ by $2$ .

$\frac{2 y}{2} = \frac{\ln (x^{2} - 6 x + 2 K)}{2}$

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Cancel the common factor of $2$ .

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Step 3.5.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{\ln (x^{2} - 6 x + 2 K)}{2}$

Step 3.5.2.1.2

Divide $y$ by $1$ .

$y = \frac{\ln (x^{2} - 6 x + 2 K)}{2}$

Step 4

Simplify the constant of integration.

$y = \frac{\ln (x^{2} - 6 x + K)}{2}$