Calculus Examples

$69 x d y + 420 y d x = 0$

Step 1

Subtract $420 y d x$ from both sides of the equation.

$69 x d y = - 420 y d x$

Step 2

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

$\frac{1}{x y} (69 x) d y = \frac{1}{x y} (- 420 y) d x$

Step 3

Simplify.

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Step 3.1

Rewrite using the commutative property of multiplication.

$69 \frac{1}{x y} x d y = \frac{1}{x y} (- 420 y) d x$

Step 3.2

Combine $69$ and $\frac{1}{x y}$ .

$\frac{69}{x y} x d y = \frac{1}{x y} (- 420 y) d x$

Step 3.3

Cancel the common factor of $x$ .

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Step 3.3.1

Factor $x$ out of $x y$ .

$\frac{69}{x (y)} x d y = \frac{1}{x y} (- 420 y) d x$

Step 3.3.2

Cancel the common factor.

$\frac{69}{x y} x d y = \frac{1}{x y} (- 420 y) d x$

Step 3.3.3

Rewrite the expression.

$\frac{69}{y} d y = \frac{1}{x y} (- 420 y) d x$

Step 3.4

Rewrite using the commutative property of multiplication.

$\frac{69}{y} d y = - 420 \frac{1}{x y} y d x$

Step 3.5

Combine $- 420$ and $\frac{1}{x y}$ .

$\frac{69}{y} d y = \frac{- 420}{x y} y d x$

Step 3.6

Cancel the common factor of $y$ .

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Step 3.6.1

Factor $y$ out of $x y$ .

$\frac{69}{y} d y = \frac{- 420}{y x} y d x$

Step 3.6.2

Cancel the common factor.

$\frac{69}{y} d y = \frac{- 420}{y x} y d x$

Step 3.6.3

Rewrite the expression.

$\frac{69}{y} d y = \frac{- 420}{x} d x$

Step 3.7

Move the negative in front of the fraction.

$\frac{69}{y} d y = - \frac{420}{x} d x$

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

$\int \frac{69}{y} d y = \int - \frac{420}{x} d x$

Step 4.2

Integrate the left side.

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Step 4.2.1

Since $69$ is constant with respect to $y$ , move $69$ out of the integral.

$69 \int \frac{1}{y} d y = \int - \frac{420}{x} d x$

Step 4.2.2

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

$69 (\ln (| y |) + C_{1}) = \int - \frac{420}{x} d x$

Step 4.2.3

Simplify.

$69 \ln (| y |) + C_{1} = \int - \frac{420}{x} d x$

Step 4.3

Integrate the right side.

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Step 4.3.1

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

$69 \ln (| y |) + C_{1} = - \int \frac{420}{x} d x$

Step 4.3.2

Since $420$ is constant with respect to $x$ , move $420$ out of the integral.

$69 \ln (| y |) + C_{1} = - (420 \int \frac{1}{x} d x)$

Step 4.3.3

Multiply $420$ by $- 1$ .

$69 \ln (| y |) + C_{1} = - 420 \int \frac{1}{x} d x$

Step 4.3.4

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

$69 \ln (| y |) + C_{1} = - 420 (\ln (| x |) + C_{2})$

Step 4.3.5

Simplify.

$69 \ln (| y |) + C_{1} = - 420 \ln (| x |) + C_{2}$

Step 4.4

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

$69 \ln (| y |) = - 420 \ln (| x |) + C$

Step 5

Solve for $y$ .

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Step 5.1

Move all the terms containing a logarithm to the left side of the equation.

$69 \ln (| y |) + 420 \ln (| x |) = C$

Step 5.2

Simplify the left side.

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Step 5.2.1

Simplify $69 \ln (| y |) + 420 \ln (| x |)$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

Simplify $69 \ln (| y |)$ by moving $69$ inside the logarithm.

$\ln ({| y |}^{69}) + 420 \ln (| x |) = C$

Step 5.2.1.1.2

Simplify $420 \ln (| x |)$ by moving $420$ inside the logarithm.

$\ln ({| y |}^{69}) + \ln ({| x |}^{420}) = C$

Step 5.2.1.1.3

Remove the absolute value in ${| x |}^{420}$ because exponentiations with even powers are always positive.

$\ln ({| y |}^{69}) + \ln (x^{420}) = C$

Step 5.2.1.2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\ln ({| y |}^{69} x^{420}) = C$

Step 5.2.1.3

Reorder factors in $\ln ({| y |}^{69} x^{420})$ .

$\ln (x^{420} {| y |}^{69}) = C$

Step 5.3

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

$e^{\ln (x^{420} {| y |}^{69})} = e^{C}$

Step 5.4

Rewrite $\ln (x^{420} {| y |}^{69}) = C$ 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 $b^{y} = x$ .

$e^{C} = x^{420} {| y |}^{69}$

Step 5.5

Solve for $y$ .

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Step 5.5.1

Rewrite the equation as $x^{420} {| y |}^{69} = e^{C}$ .

$x^{420} {| y |}^{69} = e^{C}$

Step 5.5.2

Divide each term in $x^{420} {| y |}^{69} = e^{C}$ by $x^{420}$ and simplify.

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Step 5.5.2.1

Divide each term in $x^{420} {| y |}^{69} = e^{C}$ by $x^{420}$ .

$\frac{x^{420} {| y |}^{69}}{x^{420}} = \frac{e^{C}}{x^{420}}$

Step 5.5.2.2

Simplify the left side.

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Step 5.5.2.2.1

Cancel the common factor of $x^{420}$ .

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Step 5.5.2.2.1.1

Cancel the common factor.

$\frac{x^{420} {| y |}^{69}}{x^{420}} = \frac{e^{C}}{x^{420}}$

Step 5.5.2.2.1.2

Divide ${| y |}^{69}$ by $1$ .

${| y |}^{69} = \frac{e^{C}}{x^{420}}$

Step 5.5.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$| y | = \sqrt[69]{\frac{e^{C}}{x^{420}}}$

Step 5.5.4

Simplify $\sqrt[69]{\frac{e^{C}}{x^{420}}}$ .

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Step 5.5.4.1

Rewrite $\frac{e^{C}}{x^{420}}$ as ${(\frac{1}{x^{6}})}^{69} \frac{e^{C}}{x^{6}}$ .

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Step 5.5.4.1.1

Factor the perfect power $1^{69}$ out of $e^{C}$ .

$| y | = \sqrt[69]{\frac{1^{69} e^{C}}{x^{420}}}$

Step 5.5.4.1.2

Factor the perfect power ${(x^{6})}^{69}$ out of $x^{420}$ .

$| y | = \sqrt[69]{\frac{1^{69} e^{C}}{{(x^{6})}^{69} x^{6}}}$

Step 5.5.4.1.3

Rearrange the fraction $\frac{1^{69} e^{C}}{{(x^{6})}^{69} x^{6}}$ .

$| y | = \sqrt[69]{{(\frac{1}{x^{6}})}^{69} \frac{e^{C}}{x^{6}}}$

Step 5.5.4.2

Pull terms out from under the radical.

$| y | = \frac{1}{x^{6}} \sqrt[69]{\frac{e^{C}}{x^{6}}}$

Step 5.5.4.3

Rewrite $\sqrt[69]{\frac{e^{C}}{x^{6}}}$ as $\frac{\sqrt[69]{e^{C}}}{\sqrt[69]{x^{6}}}$ .

$| y | = \frac{1}{x^{6}} \cdot \frac{\sqrt[69]{e^{C}}}{\sqrt[69]{x^{6}}}$

Step 5.5.4.4

Combine fractions.

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Step 5.5.4.4.1

Combine.

$| y | = \frac{1 \sqrt[69]{e^{C}}}{x^{6} \sqrt[69]{x^{6}}}$

Step 5.5.4.4.2

Multiply $\sqrt[69]{e^{C}}$ by $1$ .

$| y | = \frac{\sqrt[69]{e^{C}}}{x^{6} \sqrt[69]{x^{6}}}$

Step 5.5.4.5

Simplify the denominator.

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Step 5.5.4.5.1

Rewrite $x^{6}$ as ${(x^{2})}^{3}$ .

$| y | = \frac{\sqrt[69]{e^{C}}}{x^{6} \sqrt[69]{{(x^{2})}^{3}}}$

Step 5.5.4.5.2

Rewrite $\sqrt[69]{{(x^{2})}^{3}}$ as $\sqrt[23]{\sqrt[3]{{(x^{2})}^{3}}}$ .

$| y | = \frac{\sqrt[69]{e^{C}}}{x^{6} \sqrt[23]{\sqrt[3]{{(x^{2})}^{3}}}}$

Step 5.5.4.5.3

Pull terms out from under the radical, assuming real numbers.

$| y | = \frac{\sqrt[69]{e^{C}}}{x^{6} \sqrt[23]{x^{2}}}$

Step 5.5.4.6

Multiply $\frac{\sqrt[69]{e^{C}}}{x^{6} \sqrt[23]{x^{2}}}$ by $\frac{{\sqrt[23]{x^{2}}}^{22}}{{\sqrt[23]{x^{2}}}^{22}}$ .

$| y | = \frac{\sqrt[69]{e^{C}}}{x^{6} \sqrt[23]{x^{2}}} \cdot \frac{{\sqrt[23]{x^{2}}}^{22}}{{\sqrt[23]{x^{2}}}^{22}}$

Step 5.5.4.7

Combine and simplify the denominator.

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Step 5.5.4.7.1

Multiply $\frac{\sqrt[69]{e^{C}}}{x^{6} \sqrt[23]{x^{2}}}$ by $\frac{{\sqrt[23]{x^{2}}}^{22}}{{\sqrt[23]{x^{2}}}^{22}}$ .

$| y | = \frac{\sqrt[69]{e^{C}} {\sqrt[23]{x^{2}}}^{22}}{x^{6} \sqrt[23]{x^{2}} {\sqrt[23]{x^{2}}}^{22}}$

Step 5.5.4.7.2

Move $\sqrt[23]{x^{2}}$ .

$| y | = \frac{\sqrt[69]{e^{C}} {\sqrt[23]{x^{2}}}^{22}}{x^{6} (\sqrt[23]{x^{2}} {\sqrt[23]{x^{2}}}^{22})}$

Step 5.5.4.7.3

Raise $\sqrt[23]{x^{2}}$ to the power of $1$ .

$| y | = \frac{\sqrt[69]{e^{C}} {\sqrt[23]{x^{2}}}^{22}}{x^{6} ({\sqrt[23]{x^{2}}}^{1} {\sqrt[23]{x^{2}}}^{22})}$

Step 5.5.4.7.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$| y | = \frac{\sqrt[69]{e^{C}} {\sqrt[23]{x^{2}}}^{22}}{x^{6} {\sqrt[23]{x^{2}}}^{1 + 22}}$

Step 5.5.4.7.5

Add $1$ and $22$ .

$| y | = \frac{\sqrt[69]{e^{C}} {\sqrt[23]{x^{2}}}^{22}}{x^{6} {\sqrt[23]{x^{2}}}^{23}}$

Step 5.5.4.7.6

Rewrite ${\sqrt[23]{x^{2}}}^{23}$ as $x^{2}$ .

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Step 5.5.4.7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[23]{x^{2}}$ as $x^{\frac{2}{23}}$ .

$| y | = \frac{\sqrt[69]{e^{C}} {\sqrt[23]{x^{2}}}^{22}}{x^{6} {(x^{\frac{2}{23}})}^{23}}$

Step 5.5.4.7.6.2

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

$| y | = \frac{\sqrt[69]{e^{C}} {\sqrt[23]{x^{2}}}^{22}}{x^{6} x^{\frac{2}{23} \cdot 23}}$

Step 5.5.4.7.6.3

Combine $\frac{2}{23}$ and $23$ .

$| y | = \frac{\sqrt[69]{e^{C}} {\sqrt[23]{x^{2}}}^{22}}{x^{6} x^{\frac{2 \cdot 23}{23}}}$

Step 5.5.4.7.6.4

Multiply $2$ by $23$ .

$| y | = \frac{\sqrt[69]{e^{C}} {\sqrt[23]{x^{2}}}^{22}}{x^{6} x^{\frac{46}{23}}}$

Step 5.5.4.7.6.5

Cancel the common factor of $46$ and $23$ .

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Step 5.5.4.7.6.5.1

Factor $23$ out of $46$ .

$| y | = \frac{\sqrt[69]{e^{C}} {\sqrt[23]{x^{2}}}^{22}}{x^{6} x^{\frac{23 \cdot 2}{23}}}$

Step 5.5.4.7.6.5.2

Cancel the common factors.

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Step 5.5.4.7.6.5.2.1

Factor $23$ out of $23$ .

$| y | = \frac{\sqrt[69]{e^{C}} {\sqrt[23]{x^{2}}}^{22}}{x^{6} x^{\frac{23 \cdot 2}{23 (1)}}}$

Step 5.5.4.7.6.5.2.2

Cancel the common factor.

$| y | = \frac{\sqrt[69]{e^{C}} {\sqrt[23]{x^{2}}}^{22}}{x^{6} x^{\frac{23 \cdot 2}{23 \cdot 1}}}$

Step 5.5.4.7.6.5.2.3

Rewrite the expression.

$| y | = \frac{\sqrt[69]{e^{C}} {\sqrt[23]{x^{2}}}^{22}}{x^{6} x^{\frac{2}{1}}}$

Step 5.5.4.7.6.5.2.4

Divide $2$ by $1$ .

$| y | = \frac{\sqrt[69]{e^{C}} {\sqrt[23]{x^{2}}}^{22}}{x^{6} x^{2}}$

Step 5.5.4.8

Multiply $x^{6}$ by $x^{2}$ by adding the exponents.

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Step 5.5.4.8.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$| y | = \frac{\sqrt[69]{e^{C}} {\sqrt[23]{x^{2}}}^{22}}{x^{6 + 2}}$

Step 5.5.4.8.2

Add $6$ and $2$ .

$| y | = \frac{\sqrt[69]{e^{C}} {\sqrt[23]{x^{2}}}^{22}}{x^{8}}$

Step 5.5.4.9

Simplify the numerator.

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Step 5.5.4.9.1

Rewrite ${\sqrt[23]{x^{2}}}^{22}$ as $\sqrt[23]{{(x^{2})}^{22}}$ .

$| y | = \frac{\sqrt[69]{e^{C}} \sqrt[23]{{(x^{2})}^{22}}}{x^{8}}$

Step 5.5.4.9.2

Multiply the exponents in ${(x^{2})}^{22}$ .

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Step 5.5.4.9.2.1

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

$| y | = \frac{\sqrt[69]{e^{C}} \sqrt[23]{x^{2 \cdot 22}}}{x^{8}}$

Step 5.5.4.9.2.2

Multiply $2$ by $22$ .

$| y | = \frac{\sqrt[69]{e^{C}} \sqrt[23]{x^{44}}}{x^{8}}$

Step 5.5.4.9.3

Factor out $x^{23}$ .

$| y | = \frac{\sqrt[69]{e^{C}} \sqrt[23]{x^{23} x^{21}}}{x^{8}}$

Step 5.5.4.9.4

Pull terms out from under the radical.

$| y | = \frac{\sqrt[69]{e^{C}} x \sqrt[23]{x^{21}}}{x^{8}}$

Step 5.5.4.9.5

Combine exponents.

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Step 5.5.4.9.5.1

Rewrite the expression using the least common index of $69$ .

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Step 5.5.4.9.5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[23]{x^{21}}$ as $x^{\frac{21}{23}}$ .

$| y | = \frac{x (\sqrt[69]{e^{C}} x^{\frac{21}{23}})}{x^{8}}$

Step 5.5.4.9.5.1.2

Rewrite $x^{\frac{21}{23}}$ as $x^{\frac{63}{69}}$ .

$| y | = \frac{x (\sqrt[69]{e^{C}} x^{\frac{63}{69}})}{x^{8}}$

Step 5.5.4.9.5.1.3

Rewrite $x^{\frac{63}{69}}$ as $\sqrt[69]{x^{63}}$ .

$| y | = \frac{x (\sqrt[69]{e^{C}} \sqrt[69]{x^{63}})}{x^{8}}$

Step 5.5.4.9.5.2

Combine using the product rule for radicals.

$| y | = \frac{x \sqrt[69]{e^{C} x^{63}}}{x^{8}}$

Step 5.5.4.10

Reduce the expression by cancelling the common factors.

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Step 5.5.4.10.1

Cancel the common factor of $x$ and $x^{8}$ .

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Step 5.5.4.10.1.1

Factor $x$ out of $x \sqrt[69]{e^{C} x^{63}}$ .

$| y | = \frac{x \sqrt[69]{e^{C} x^{63}}}{x^{8}}$

Step 5.5.4.10.1.2

Cancel the common factors.

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Step 5.5.4.10.1.2.1

Factor $x$ out of $x^{8}$ .

$| y | = \frac{x \sqrt[69]{e^{C} x^{63}}}{x \cdot x^{7}}$

Step 5.5.4.10.1.2.2

Cancel the common factor.

$| y | = \frac{x \sqrt[69]{e^{C} x^{63}}}{x \cdot x^{7}}$

Step 5.5.4.10.1.2.3

Rewrite the expression.

$| y | = \frac{\sqrt[69]{e^{C} x^{63}}}{x^{7}}$

Step 5.5.4.10.2

Reorder factors in $\frac{\sqrt[69]{e^{C} x^{63}}}{x^{7}}$ .

$| y | = \frac{\sqrt[69]{x^{63} e^{C}}}{x^{7}}$

Step 5.5.5

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

$y = \pm \frac{\sqrt[69]{x^{63} e^{C}}}{x^{7}}$

Step 6

Simplify the constant of integration.

$y = \pm \frac{\sqrt[69]{x^{63} C}}{x^{7}}$