Calculus Examples

$2 \frac{d y}{d x} = (10 + 4 \sin (t)) y$

Step 1

Separate the variables.

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Step 1.1

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

$\frac{1}{y} (2 \frac{d y}{d x}) = \frac{1}{y} ((10 + 4 \sin (t)) y)$

Step 1.2

Cancel the common factor of $y$ .

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Step 1.2.1

Factor $y$ out of $(10 + 4 \sin (t)) y$ .

$\frac{1}{y} (2 \frac{d y}{d x}) = \frac{1}{y} (y (10 + 4 \sin (t)))$

Step 1.2.2

Cancel the common factor.

$\frac{1}{y} (2 \frac{d y}{d x}) = \frac{1}{y} (y (10 + 4 \sin (t)))$

Step 1.2.3

Rewrite the expression.

$\frac{1}{y} (2 \frac{d y}{d x}) = 10 + 4 \sin (t)$

Step 1.3

Remove unnecessary parentheses.

$\frac{1}{y} \cdot 2 \frac{d y}{d x} = 10 + 4 \sin (t)$

Step 1.4

Rewrite the equation.

$\frac{1}{y} \cdot 2 d y = (10 + 4 \sin (t)) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{y} \cdot 2 d y = \int 10 + 4 \sin (t) d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

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

$\int \frac{2}{y} d y = \int 10 + 4 \sin (t) d x$

Step 2.2.2

Since $2$ is constant with respect to $y$ , move $2$ out of the integral.

$2 \int \frac{1}{y} d y = \int 10 + 4 \sin (t) d x$

Step 2.2.3

The integral of $\frac{1}{y}$ with respect to $y$ is $\ln (| y |)$ .

$2 (\ln (| y |) + C_{1}) = \int 10 + 4 \sin (t) d x$

Step 2.2.4

Simplify.

$2 \ln (| y |) + C_{1} = \int 10 + 4 \sin (t) d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Apply the constant rule.

$2 \ln (| y |) + C_{1} = (10 + 4 \sin (t)) x + C_{2}$

Step 2.3.2

Reorder terms.

$2 \ln (| y |) + C_{1} = x (10 + 4 \sin (t)) + C_{2}$

Step 2.4

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

$2 \ln (| y |) = x (10 + 4 \sin (t)) + C$

Step 3

Solve for $y$ .

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Step 3.1

Divide each term in $2 \ln (| y |) = x (10 + 4 \sin (t)) + C$ by $2$ and simplify.

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Step 3.1.1

Divide each term in $2 \ln (| y |) = x (10 + 4 \sin (t)) + C$ by $2$ .

$\frac{2 \ln (| y |)}{2} = \frac{x (10 + 4 \sin (t))}{2} + \frac{C}{2}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $2$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{2 \ln (| y |)}{2} = \frac{x (10 + 4 \sin (t))}{2} + \frac{C}{2}$

Step 3.1.2.1.2

Divide $\ln (| y |)$ by $1$ .

$\ln (| y |) = \frac{x (10 + 4 \sin (t))}{2} + \frac{C}{2}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Simplify each term.

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Step 3.1.3.1.1

Cancel the common factor of $10 + 4 \sin (t)$ and $2$ .

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Step 3.1.3.1.1.1

Factor $2$ out of $x (10 + 4 \sin (t))$ .

$\ln (| y |) = \frac{2 (x (5 + 2 \sin (t)))}{2} + \frac{C}{2}$

Step 3.1.3.1.1.2

Cancel the common factors.

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Step 3.1.3.1.1.2.1

Factor $2$ out of $2$ .

$\ln (| y |) = \frac{2 (x (5 + 2 \sin (t)))}{2 (1)} + \frac{C}{2}$

Step 3.1.3.1.1.2.2

Cancel the common factor.

$\ln (| y |) = \frac{2 (x (5 + 2 \sin (t)))}{2 \cdot 1} + \frac{C}{2}$

Step 3.1.3.1.1.2.3

Rewrite the expression.

$\ln (| y |) = \frac{x (5 + 2 \sin (t))}{1} + \frac{C}{2}$

Step 3.1.3.1.1.2.4

Divide $x (5 + 2 \sin (t))$ by $1$ .

$\ln (| y |) = x (5 + 2 \sin (t)) + \frac{C}{2}$

Step 3.1.3.1.2

Apply the distributive property.

$\ln (| y |) = x \cdot 5 + x (2 \sin (t)) + \frac{C}{2}$

Step 3.1.3.1.3

Move $5$ to the left of $x$ .

$\ln (| y |) = 5 \cdot x + x (2 \sin (t)) + \frac{C}{2}$

Step 3.1.3.1.4

Rewrite using the commutative property of multiplication.

$\ln (| y |) = 5 x + 2 x \sin (t) + \frac{C}{2}$

Step 3.2

To solve for $y$ , rewrite the equation using properties of logarithms.

$e^{\ln (| y |)} = e^{5 x + 2 x \sin (t) + \frac{C}{2}}$

Step 3.3

Rewrite $\ln (| y |) = 5 x + 2 x \sin (t) + \frac{C}{2}$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{5 x + 2 x \sin (t) + \frac{C}{2}} = | y |$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Rewrite the equation as $| y | = e^{5 x + 2 x \sin (t) + \frac{C}{2}}$ .

$| y | = e^{5 x + 2 x \sin (t) + \frac{C}{2}}$

Step 3.4.2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$y = \pm e^{5 x + 2 x \sin (t) + \frac{C}{2}}$

Step 4

Group the constant terms together.

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Step 4.1

Simplify the constant of integration.

$y = \pm e^{5 x + 2 x \sin (t) + C}$

Step 4.2

Rewrite $e^{5 x + 2 x \sin (t) + C}$ as $e^{5 x + 2 x \sin (t)} e^{C}$ .

$y = \pm e^{5 x + 2 x \sin (t)} e^{C}$

Step 4.3

Reorder $e^{5 x + 2 x \sin (t)}$ and $e^{C}$ .

$y = \pm e^{C} e^{5 x + 2 x \sin (t)}$

Step 4.4

Combine constants with the plus or minus.

$y = C e^{5 x + 2 x \sin (t)}$