Calculus Examples

$y d x + 3 x d y = 0$

Step 1

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

$3 x d y = - y d x$

Step 2

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

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

Step 3

Simplify.

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Step 3.1

Rewrite using the commutative property of multiplication.

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

Step 3.2

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

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

Step 3.3

Cancel the common factor of $x$ .

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Step 3.3.1

Factor $x$ out of $x y$ .

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

Step 3.3.2

Cancel the common factor.

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

Step 3.3.3

Rewrite the expression.

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

Step 3.4

Rewrite using the commutative property of multiplication.

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

Step 3.5

Cancel the common factor of $y$ .

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Step 3.5.1

Move the leading negative in $- \frac{1}{x y}$ into the numerator.

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

Step 3.5.2

Factor $y$ out of $x y$ .

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

Step 3.5.3

Cancel the common factor.

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

Step 3.5.4

Rewrite the expression.

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

Step 3.6

Move the negative in front of the fraction.

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

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

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

Step 4.2

Integrate the left side.

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Step 4.2.1

Since $3$ is constant with respect to $y$ , move $3$ out of the integral.

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

Step 4.2.2

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

$3 (\ln (| y |) + C_{1}) = \int - \frac{1}{x} d x$

Step 4.2.3

Simplify.

$3 \ln (| y |) + C_{1} = \int - \frac{1}{x} d x$

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.

$3 \ln (| y |) + C_{1} = - \int \frac{1}{x} d x$

Step 4.3.2

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

$3 \ln (| y |) + C_{1} = - (\ln (| x |) + C_{2})$

Step 4.3.3

Simplify.

$3 \ln (| y |) + C_{1} = - \ln (| x |) + C_{2}$

Step 4.4

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

$3 \ln (| y |) = - \ln (| x |) + C$

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.

$3 \ln (| y |) + \ln (| x |) = C$

Step 5.2

Simplify the left side.

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Step 5.2.1

Simplify $3 \ln (| y |) + \ln (| x |)$ .

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Step 5.2.1.1

Simplify $3 \ln (| y |)$ by moving $3$ inside the logarithm.

$\ln ({| y |}^{3}) + \ln (| x |) = C$

Step 5.2.1.2

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

$\ln ({| y |}^{3} | x |) = C$

Step 5.3

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

$e^{\ln ({| y |}^{3} | x |)} = e^{C}$

Step 5.4

Rewrite $\ln ({| y |}^{3} | x |) = C$ 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 $b^{y} = x$ .

$e^{C} = {| y |}^{3} | x |$

Step 5.5

Solve for $y$ .

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Step 5.5.1

Rewrite the equation as ${| y |}^{3} | x | = e^{C}$ .

${| y |}^{3} | x | = e^{C}$

Step 5.5.2

Divide each term in ${| y |}^{3} | x | = e^{C}$ by $| x |$ and simplify.

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Step 5.5.2.1

Divide each term in ${| y |}^{3} | x | = e^{C}$ by $| x |$ .

$\frac{{| y |}^{3} | x |}{| x |} = \frac{e^{C}}{| x |}$

Step 5.5.2.2

Simplify the left side.

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Step 5.5.2.2.1

Cancel the common factor of $| x |$ .

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Step 5.5.2.2.1.1

Cancel the common factor.

$\frac{{| y |}^{3} | x |}{| x |} = \frac{e^{C}}{| x |}$

Step 5.5.2.2.1.2

Divide ${| y |}^{3}$ by $1$ .

${| y |}^{3} = \frac{e^{C}}{| x |}$

Step 5.5.3

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

$| y | = \sqrt[3]{\frac{e^{C}}{| x |}}$

Step 5.5.4

Simplify $\sqrt[3]{\frac{e^{C}}{| x |}}$ .

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Step 5.5.4.1

Rewrite $\sqrt[3]{\frac{e^{C}}{| x |}}$ as $\frac{\sqrt[3]{e^{C}}}{\sqrt[3]{| x |}}$ .

$| y | = \frac{\sqrt[3]{e^{C}}}{\sqrt[3]{| x |}}$

Step 5.5.4.2

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

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

Step 5.5.4.3

Combine and simplify the denominator.

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Step 5.5.4.3.1

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

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

Step 5.5.4.3.2

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

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

Step 5.5.4.3.3

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

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

Step 5.5.4.3.4

Add $1$ and $2$ .

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

Step 5.5.4.3.5

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

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Step 5.5.4.3.5.1

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

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

Step 5.5.4.3.5.2

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

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

Step 5.5.4.3.5.3

Combine $\frac{1}{3}$ and $3$ .

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

Step 5.5.4.3.5.4

Cancel the common factor of $3$ .

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Step 5.5.4.3.5.4.1

Cancel the common factor.

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

Step 5.5.4.3.5.4.2

Rewrite the expression.

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

Step 5.5.4.3.5.5

Simplify.

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

Step 5.5.4.4

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

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

Step 5.5.4.5

Combine using the product rule for radicals.

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

Step 5.5.4.6

Reorder factors in $\frac{\sqrt[3]{e^{C} {| x |}^{2}}}{| x |}$ .

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

Step 5.5.5

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

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

Step 5.5.6

Remove the absolute value in ${| x |}^{2}$ because exponentiations with even powers are always positive.

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

Step 6

Simplify the constant of integration.

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