Calculus Examples

$8 x y^{3} d x + 12 x^{2} y^{2} d y = 0$

Step 1

Subtract $8 x y^{3} d x$ from both sides of the equation.

$12 x^{2} y^{2} d y = - 8 x y^{3} d x$

Step 2

Multiply both sides by $\frac{1}{x^{2} y^{3}}$ .

$\frac{1}{x^{2} y^{3}} (12 x^{2} y^{2}) d y = \frac{1}{x^{2} y^{3}} (- 8 x y^{3}) d x$

Step 3

Simplify.

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Step 3.1

Rewrite using the commutative property of multiplication.

$12 \frac{1}{x^{2} y^{3}} (x^{2} y^{2}) d y = \frac{1}{x^{2} y^{3}} (- 8 x y^{3}) d x$

Step 3.2

Combine $12$ and $\frac{1}{x^{2} y^{3}}$ .

$\frac{12}{x^{2} y^{3}} (x^{2} y^{2}) d y = \frac{1}{x^{2} y^{3}} (- 8 x y^{3}) d x$

Step 3.3

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

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Step 3.3.1

Factor $x^{2} y^{2}$ out of $x^{2} y^{3}$ .

$\frac{12}{x^{2} y^{2} (y)} (x^{2} y^{2}) d y = \frac{1}{x^{2} y^{3}} (- 8 x y^{3}) d x$

Step 3.3.2

Cancel the common factor.

$\frac{12}{x^{2} y^{2} y} (x^{2} y^{2}) d y = \frac{1}{x^{2} y^{3}} (- 8 x y^{3}) d x$

Step 3.3.3

Rewrite the expression.

$\frac{12}{y} d y = \frac{1}{x^{2} y^{3}} (- 8 x y^{3}) d x$

Step 3.4

Rewrite using the commutative property of multiplication.

$\frac{12}{y} d y = - 8 \frac{1}{x^{2} y^{3}} (x y^{3}) d x$

Step 3.5

Combine $- 8$ and $\frac{1}{x^{2} y^{3}}$ .

$\frac{12}{y} d y = \frac{- 8}{x^{2} y^{3}} (x y^{3}) d x$

Step 3.6

Cancel the common factor of $x y^{3}$ .

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Step 3.6.1

Factor $x y^{3}$ out of $x^{2} y^{3}$ .

$\frac{12}{y} d y = \frac{- 8}{x y^{3} (x)} (x y^{3}) d x$

Step 3.6.2

Cancel the common factor.

$\frac{12}{y} d y = \frac{- 8}{x y^{3} x} (x y^{3}) d x$

Step 3.6.3

Rewrite the expression.

$\frac{12}{y} d y = \frac{- 8}{x} d x$

Step 3.7

Move the negative in front of the fraction.

$\frac{12}{y} d y = - \frac{8}{x} d x$

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

$\int \frac{12}{y} d y = \int - \frac{8}{x} d x$

Step 4.2

Integrate the left side.

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Step 4.2.1

Since $12$ is constant with respect to $y$ , move $12$ out of the integral.

$12 \int \frac{1}{y} d y = \int - \frac{8}{x} d x$

Step 4.2.2

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

$12 (\ln (| y |) + C_{1}) = \int - \frac{8}{x} d x$

Step 4.2.3

Simplify.

$12 \ln (| y |) + C_{1} = \int - \frac{8}{x} d x$

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.

$12 \ln (| y |) + C_{1} = - \int \frac{8}{x} d x$

Step 4.3.2

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$12 \ln (| y |) + C_{1} = - (8 \int \frac{1}{x} d x)$

Step 4.3.3

Multiply $8$ by $- 1$ .

$12 \ln (| y |) + C_{1} = - 8 \int \frac{1}{x} d x$

Step 4.3.4

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

$12 \ln (| y |) + C_{1} = - 8 (\ln (| x |) + C_{2})$

Step 4.3.5

Simplify.

$12 \ln (| y |) + C_{1} = - 8 \ln (| x |) + C_{2}$

Step 4.4

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

$12 \ln (| y |) = - 8 \ln (| x |) + C$

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.

$12 \ln (| y |) + 8 \ln (| x |) = C$

Step 5.2

Simplify the left side.

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Step 5.2.1

Simplify $12 \ln (| y |) + 8 \ln (| x |)$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

Simplify $12 \ln (| y |)$ by moving $12$ inside the logarithm.

$\ln ({| y |}^{12}) + 8 \ln (| x |) = C$

Step 5.2.1.1.2

Remove the absolute value in ${| y |}^{12}$ because exponentiations with even powers are always positive.

$\ln (y^{12}) + 8 \ln (| x |) = C$

Step 5.2.1.1.3

Simplify $8 \ln (| x |)$ by moving $8$ inside the logarithm.

$\ln (y^{12}) + \ln ({| x |}^{8}) = C$

Step 5.2.1.1.4

Remove the absolute value in ${| x |}^{8}$ because exponentiations with even powers are always positive.

$\ln (y^{12}) + \ln (x^{8}) = C$

Step 5.2.1.2

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

$\ln (y^{12} x^{8}) = C$

Step 5.3

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

$e^{\ln (y^{12} x^{8})} = e^{C}$

Step 5.4

Rewrite $\ln (y^{12} x^{8}) = C$ 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 $b^{y} = x$ .

$e^{C} = y^{12} x^{8}$

Step 5.5

Solve for $y$ .

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Step 5.5.1

Rewrite the equation as $y^{12} x^{8} = e^{C}$ .

$y^{12} x^{8} = e^{C}$

Step 5.5.2

Divide each term in $y^{12} x^{8} = e^{C}$ by $x^{8}$ and simplify.

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Step 5.5.2.1

Divide each term in $y^{12} x^{8} = e^{C}$ by $x^{8}$ .

$\frac{y^{12} x^{8}}{x^{8}} = \frac{e^{C}}{x^{8}}$

Step 5.5.2.2

Simplify the left side.

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Step 5.5.2.2.1

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

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Step 5.5.2.2.1.1

Cancel the common factor.

$\frac{y^{12} x^{8}}{x^{8}} = \frac{e^{C}}{x^{8}}$

Step 5.5.2.2.1.2

Divide $y^{12}$ by $1$ .

$y^{12} = \frac{e^{C}}{x^{8}}$

Step 5.5.3

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

$y = \pm \sqrt[12]{\frac{e^{C}}{x^{8}}}$

Step 5.5.4

Simplify $\pm \sqrt[12]{\frac{e^{C}}{x^{8}}}$ .

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Step 5.5.4.1

Rewrite $\sqrt[12]{\frac{e^{C}}{x^{8}}}$ as $\frac{\sqrt[12]{e^{C}}}{\sqrt[12]{x^{8}}}$ .

$y = \pm \frac{\sqrt[12]{e^{C}}}{\sqrt[12]{x^{8}}}$

Step 5.5.4.2

Simplify the denominator.

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Step 5.5.4.2.1

Rewrite $x^{8}$ as ${(x^{2})}^{4}$ .

$y = \pm \frac{\sqrt[12]{e^{C}}}{\sqrt[12]{{(x^{2})}^{4}}}$

Step 5.5.4.2.2

Rewrite $\sqrt[12]{{(x^{2})}^{4}}$ as $\sqrt[3]{\sqrt[4]{{(x^{2})}^{4}}}$ .

$y = \pm \frac{\sqrt[12]{e^{C}}}{\sqrt[3]{\sqrt[4]{{(x^{2})}^{4}}}}$

Step 5.5.4.2.3

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

$y = \pm \frac{\sqrt[12]{e^{C}}}{\sqrt[3]{x^{2}}}$

Step 5.5.4.3

Multiply $\frac{\sqrt[12]{e^{C}}}{\sqrt[3]{x^{2}}}$ by $\frac{{\sqrt[3]{x^{2}}}^{2}}{{\sqrt[3]{x^{2}}}^{2}}$ .

$y = \pm \frac{\sqrt[12]{e^{C}}}{\sqrt[3]{x^{2}}} \cdot \frac{{\sqrt[3]{x^{2}}}^{2}}{{\sqrt[3]{x^{2}}}^{2}}$

Step 5.5.4.4

Combine and simplify the denominator.

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Step 5.5.4.4.1

Multiply $\frac{\sqrt[12]{e^{C}}}{\sqrt[3]{x^{2}}}$ by $\frac{{\sqrt[3]{x^{2}}}^{2}}{{\sqrt[3]{x^{2}}}^{2}}$ .

$y = \pm \frac{\sqrt[12]{e^{C}} {\sqrt[3]{x^{2}}}^{2}}{\sqrt[3]{x^{2}} {\sqrt[3]{x^{2}}}^{2}}$

Step 5.5.4.4.2

Raise $\sqrt[3]{x^{2}}$ to the power of $1$ .

$y = \pm \frac{\sqrt[12]{e^{C}} {\sqrt[3]{x^{2}}}^{2}}{{\sqrt[3]{x^{2}}}^{1} {\sqrt[3]{x^{2}}}^{2}}$

Step 5.5.4.4.3

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

$y = \pm \frac{\sqrt[12]{e^{C}} {\sqrt[3]{x^{2}}}^{2}}{{\sqrt[3]{x^{2}}}^{1 + 2}}$

Step 5.5.4.4.4

Add $1$ and $2$ .

$y = \pm \frac{\sqrt[12]{e^{C}} {\sqrt[3]{x^{2}}}^{2}}{{\sqrt[3]{x^{2}}}^{3}}$

Step 5.5.4.4.5

Rewrite ${\sqrt[3]{x^{2}}}^{3}$ as $x^{2}$ .

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Step 5.5.4.4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

$y = \pm \frac{\sqrt[12]{e^{C}} {\sqrt[3]{x^{2}}}^{2}}{{(x^{\frac{2}{3}})}^{3}}$

Step 5.5.4.4.5.2

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

$y = \pm \frac{\sqrt[12]{e^{C}} {\sqrt[3]{x^{2}}}^{2}}{x^{\frac{2}{3} \cdot 3}}$

Step 5.5.4.4.5.3

Combine $\frac{2}{3}$ and $3$ .

$y = \pm \frac{\sqrt[12]{e^{C}} {\sqrt[3]{x^{2}}}^{2}}{x^{\frac{2 \cdot 3}{3}}}$

Step 5.5.4.4.5.4

Multiply $2$ by $3$ .

$y = \pm \frac{\sqrt[12]{e^{C}} {\sqrt[3]{x^{2}}}^{2}}{x^{\frac{6}{3}}}$

Step 5.5.4.4.5.5

Cancel the common factor of $6$ and $3$ .

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Step 5.5.4.4.5.5.1

Factor $3$ out of $6$ .

$y = \pm \frac{\sqrt[12]{e^{C}} {\sqrt[3]{x^{2}}}^{2}}{x^{\frac{3 \cdot 2}{3}}}$

Step 5.5.4.4.5.5.2

Cancel the common factors.

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Step 5.5.4.4.5.5.2.1

Factor $3$ out of $3$ .

$y = \pm \frac{\sqrt[12]{e^{C}} {\sqrt[3]{x^{2}}}^{2}}{x^{\frac{3 \cdot 2}{3 (1)}}}$

Step 5.5.4.4.5.5.2.2

Cancel the common factor.

$y = \pm \frac{\sqrt[12]{e^{C}} {\sqrt[3]{x^{2}}}^{2}}{x^{\frac{3 \cdot 2}{3 \cdot 1}}}$

Step 5.5.4.4.5.5.2.3

Rewrite the expression.

$y = \pm \frac{\sqrt[12]{e^{C}} {\sqrt[3]{x^{2}}}^{2}}{x^{\frac{2}{1}}}$

Step 5.5.4.4.5.5.2.4

Divide $2$ by $1$ .

$y = \pm \frac{\sqrt[12]{e^{C}} {\sqrt[3]{x^{2}}}^{2}}{x^{2}}$

Step 5.5.4.5

Simplify the numerator.

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Step 5.5.4.5.1

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

$y = \pm \frac{\sqrt[12]{e^{C}} \sqrt[3]{{(x^{2})}^{2}}}{x^{2}}$

Step 5.5.4.5.2

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

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Step 5.5.4.5.2.1

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

$y = \pm \frac{\sqrt[12]{e^{C}} \sqrt[3]{x^{2 \cdot 2}}}{x^{2}}$

Step 5.5.4.5.2.2

Multiply $2$ by $2$ .

$y = \pm \frac{\sqrt[12]{e^{C}} \sqrt[3]{x^{4}}}{x^{2}}$

Step 5.5.4.5.3

Factor out $x^{3}$ .

$y = \pm \frac{\sqrt[12]{e^{C}} \sqrt[3]{x^{3} x}}{x^{2}}$

Step 5.5.4.5.4

Pull terms out from under the radical.

$y = \pm \frac{\sqrt[12]{e^{C}} x \sqrt[3]{x}}{x^{2}}$

Step 5.5.4.5.5

Combine exponents.

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Step 5.5.4.5.5.1

Rewrite the expression using the least common index of $12$ .

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Step 5.5.4.5.5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

$y = \pm \frac{x (\sqrt[12]{e^{C}} x^{\frac{1}{3}})}{x^{2}}$

Step 5.5.4.5.5.1.2

Rewrite $x^{\frac{1}{3}}$ as $x^{\frac{4}{12}}$ .

$y = \pm \frac{x (\sqrt[12]{e^{C}} x^{\frac{4}{12}})}{x^{2}}$

Step 5.5.4.5.5.1.3

Rewrite $x^{\frac{4}{12}}$ as $\sqrt[12]{x^{4}}$ .

$y = \pm \frac{x (\sqrt[12]{e^{C}} \sqrt[12]{x^{4}})}{x^{2}}$

Step 5.5.4.5.5.2

Combine using the product rule for radicals.

$y = \pm \frac{x \sqrt[12]{e^{C} x^{4}}}{x^{2}}$

Step 5.5.4.6

Reduce the expression by cancelling the common factors.

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Step 5.5.4.6.1

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

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Step 5.5.4.6.1.1

Factor $x$ out of $x \sqrt[12]{e^{C} x^{4}}$ .

$y = \pm \frac{x \sqrt[12]{e^{C} x^{4}}}{x^{2}}$

Step 5.5.4.6.1.2

Cancel the common factors.

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Step 5.5.4.6.1.2.1

Factor $x$ out of $x^{2}$ .

$y = \pm \frac{x \sqrt[12]{e^{C} x^{4}}}{x \cdot x}$

Step 5.5.4.6.1.2.2

Cancel the common factor.

$y = \pm \frac{x \sqrt[12]{e^{C} x^{4}}}{x \cdot x}$

Step 5.5.4.6.1.2.3

Rewrite the expression.

$y = \pm \frac{\sqrt[12]{e^{C} x^{4}}}{x}$

Step 5.5.4.6.2

Reorder factors in $\pm \frac{\sqrt[12]{e^{C} x^{4}}}{x}$ .

$y = \pm \frac{\sqrt[12]{x^{4} e^{C}}}{x}$

Step 5.5.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.5.5.1

First, use the positive value of the $\pm$ to find the first solution.