Calculus Examples

$\frac{d y}{d t} - 42 t^{6} e^{- y} = 0$

Step 1

Separate the variables.

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Step 1.1

Add $42 t^{6} e^{- y}$ to both sides of the equation.

$\frac{d y}{d t} = 42 t^{6} e^{- y}$

Step 1.2

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

$\frac{1}{e^{- y}} \frac{d y}{d t} = \frac{1}{e^{- y}} (42 t^{6} e^{- y})$

Step 1.3

Simplify.

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Step 1.3.1

Rewrite using the commutative property of multiplication.

$\frac{1}{e^{- y}} \frac{d y}{d t} = 42 \frac{1}{e^{- y}} (t^{6} e^{- y})$

Step 1.3.2

Combine $42$ and $\frac{1}{e^{- y}}$ .

$\frac{1}{e^{- y}} \frac{d y}{d t} = \frac{42}{e^{- y}} (t^{6} e^{- y})$

Step 1.3.3

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

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Step 1.3.3.1

Factor $e^{- y}$ out of $t^{6} e^{- y}$ .

$\frac{1}{e^{- y}} \frac{d y}{d t} = \frac{42}{e^{- y}} (e^{- y} t^{6})$

Step 1.3.3.2

Cancel the common factor.

$\frac{1}{e^{- y}} \frac{d y}{d t} = \frac{42}{e^{- y}} (e^{- y} t^{6})$

Step 1.3.3.3

Rewrite the expression.

$\frac{1}{e^{- y}} \frac{d y}{d t} = 42 t^{6}$

Step 1.4

Rewrite the equation.

$\frac{1}{e^{- y}} d y = 42 t^{6} d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{e^{- y}} d y = \int 42 t^{6} d t$

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.

$\int 1 {(e^{- y})}^{- 1} d y = \int 42 t^{6} d t$

Step 2.2.1.2

Simplify.

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Step 2.2.1.2.1

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

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Step 2.2.1.2.1.1

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

$\int 1 e^{- y \cdot - 1} d y = \int 42 t^{6} d t$

Step 2.2.1.2.1.2

Multiply $- y \cdot - 1$ .

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Step 2.2.1.2.1.2.1

Multiply $- 1$ by $- 1$ .

$\int 1 e^{1 y} d y = \int 42 t^{6} d t$

Step 2.2.1.2.1.2.2

Multiply $y$ by $1$ .

$\int 1 e^{y} d y = \int 42 t^{6} d t$

Step 2.2.1.2.2

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

$\int e^{y} d y = \int 42 t^{6} d t$

Step 2.2.2

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

$e^{y} + C_{1} = \int 42 t^{6} d t$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $42$ is constant with respect to $t$ , move $42$ out of the integral.

$e^{y} + C_{1} = 42 \int t^{6} d t$

Step 2.3.2

By the Power Rule, the integral of $t^{6}$ with respect to $t$ is $\frac{1}{7} t^{7}$ .

$e^{y} + C_{1} = 42 (\frac{1}{7} t^{7} + C_{2})$

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite $42 (\frac{1}{7} t^{7} + C_{2})$ as $42 (\frac{1}{7}) t^{7} + C_{2}$ .

$e^{y} + C_{1} = 42 (\frac{1}{7}) t^{7} + C_{2}$

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

Combine $42$ and $\frac{1}{7}$ .

$e^{y} + C_{1} = \frac{42}{7} t^{7} + C_{2}$

Step 2.3.3.2.2

Cancel the common factor of $42$ and $7$ .

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Step 2.3.3.2.2.1

Factor $7$ out of $42$ .

$e^{y} + C_{1} = \frac{7 \cdot 6}{7} t^{7} + C_{2}$

Step 2.3.3.2.2.2

Cancel the common factors.

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Step 2.3.3.2.2.2.1

Factor $7$ out of $7$ .

$e^{y} + C_{1} = \frac{7 \cdot 6}{7 (1)} t^{7} + C_{2}$

Step 2.3.3.2.2.2.2

Cancel the common factor.

$e^{y} + C_{1} = \frac{7 \cdot 6}{7 \cdot 1} t^{7} + C_{2}$

Step 2.3.3.2.2.2.3

Rewrite the expression.

$e^{y} + C_{1} = \frac{6}{1} t^{7} + C_{2}$

Step 2.3.3.2.2.2.4

Divide $6$ by $1$ .

$e^{y} + C_{1} = 6 t^{7} + C_{2}$

Step 2.4

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

$e^{y} = 6 t^{7} + K$

Step 3

Solve for $y$ .

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Step 3.1

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

$\ln (e^{y}) = \ln (6 t^{7} + K)$

Step 3.2

Expand the left side.

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Step 3.2.1

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

$y \ln (e) = \ln (6 t^{7} + K)$

Step 3.2.2

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

$y \cdot 1 = \ln (6 t^{7} + K)$

Step 3.2.3

Multiply $y$ by $1$ .

$y = \ln (6 t^{7} + K)$