Calculus Examples

$6 y^{2} d x = 9 x y d y$

Step 1

Rewrite the equation.

$9 x y d y = 6 y^{2} d x$

Step 2

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

$\frac{1}{x y^{2}} (9 x y) d y = \frac{1}{x y^{2}} (6 y^{2}) d x$

Step 3

Simplify.

Tap for more steps...

Step 3.1

Rewrite using the commutative property of multiplication.

$9 \frac{1}{x y^{2}} (x y) d y = \frac{1}{x y^{2}} (6 y^{2}) d x$

Step 3.2

Combine $9$ and $\frac{1}{x y^{2}}$ .

$\frac{9}{x y^{2}} (x y) d y = \frac{1}{x y^{2}} (6 y^{2}) d x$

Step 3.3

Cancel the common factor of $x y$ .

Tap for more steps...

Step 3.3.1

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

$\frac{9}{x y (y)} (x y) d y = \frac{1}{x y^{2}} (6 y^{2}) d x$

Step 3.3.2

Cancel the common factor.

$\frac{9}{x y \cdot y} (x y) d y = \frac{1}{x y^{2}} (6 y^{2}) d x$

Step 3.3.3

Rewrite the expression.

$\frac{9}{y} d y = \frac{1}{x y^{2}} (6 y^{2}) d x$

Step 3.4

Rewrite using the commutative property of multiplication.

$\frac{9}{y} d y = 6 \frac{1}{x y^{2}} y^{2} d x$

Step 3.5

Combine $6$ and $\frac{1}{x y^{2}}$ .

$\frac{9}{y} d y = \frac{6}{x y^{2}} y^{2} d x$

Step 3.6

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

Tap for more steps...

Step 3.6.1

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

$\frac{9}{y} d y = \frac{6}{y^{2} x} y^{2} d x$

Step 3.6.2

Cancel the common factor.

$\frac{9}{y} d y = \frac{6}{y^{2} x} y^{2} d x$

Step 3.6.3

Rewrite the expression.

$\frac{9}{y} d y = \frac{6}{x} d x$

Step 4

Integrate both sides.

Tap for more steps...

Step 4.1

Set up an integral on each side.

$\int \frac{9}{y} d y = \int \frac{6}{x} d x$

Step 4.2

Integrate the left side.

Tap for more steps...

Step 4.2.1

Since $9$ is constant with respect to $y$ , move $9$ out of the integral.

$9 \int \frac{1}{y} d y = \int \frac{6}{x} d x$

Step 4.2.2

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

$9 (\ln (| y |) + C_{1}) = \int \frac{6}{x} d x$

Step 4.2.3

Simplify.

$9 \ln (| y |) + C_{1} = \int \frac{6}{x} d x$

Step 4.3

Integrate the right side.

Tap for more steps...

Step 4.3.1

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$9 \ln (| y |) + C_{1} = 6 \int \frac{1}{x} d x$

Step 4.3.2

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

$9 \ln (| y |) + C_{1} = 6 (\ln (| x |) + C_{2})$

Step 4.3.3

Simplify.

$9 \ln (| y |) + C_{1} = 6 \ln (| x |) + C_{2}$

Step 4.4

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

$9 \ln (| y |) = 6 \ln (| x |) + C$

Step 5

Solve for $y$ .

Tap for more steps...

Step 5.1

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

$9 \ln (| y |) - 6 \ln (| x |) = C$

Step 5.2

Simplify the left side.

Tap for more steps...

Step 5.2.1

Simplify $9 \ln (| y |) - 6 \ln (| x |)$ .

Tap for more steps...

Step 5.2.1.1

Simplify each term.

Tap for more steps...

Step 5.2.1.1.1

Simplify $9 \ln (| y |)$ by moving $9$ inside the logarithm.

$\ln ({| y |}^{9}) - 6 \ln (| x |) = C$

Step 5.2.1.1.2

Simplify $- 6 \ln (| x |)$ by moving $6$ inside the logarithm.

$\ln ({| y |}^{9}) - \ln ({| x |}^{6}) = C$

Step 5.2.1.1.3

Remove the absolute value in ${| x |}^{6}$ because exponentiations with even powers are always positive.

$\ln ({| y |}^{9}) - \ln (x^{6}) = C$

Step 5.2.1.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{{| y |}^{9}}{x^{6}}) = C$

Step 5.3

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

$e^{\ln (\frac{{| y |}^{9}}{x^{6}})} = e^{C}$

Step 5.4

Rewrite $\ln (\frac{{| y |}^{9}}{x^{6}}) = C$ 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 $b^{y} = x$ .

$e^{C} = \frac{{| y |}^{9}}{x^{6}}$

Step 5.5

Solve for $y$ .

Tap for more steps...

Step 5.5.1

Rewrite the equation as $\frac{{| y |}^{9}}{x^{6}} = e^{C}$ .

$\frac{{| y |}^{9}}{x^{6}} = e^{C}$

Step 5.5.2

Multiply both sides by $x^{6}$ .

$\frac{{| y |}^{9}}{x^{6}} x^{6} = e^{C} x^{6}$

Step 5.5.3

Simplify.

Tap for more steps...

Step 5.5.3.1

Simplify the left side.

Tap for more steps...

Step 5.5.3.1.1

Cancel the common factor of $x^{6}$ .

Tap for more steps...

Step 5.5.3.1.1.1

Cancel the common factor.

$\frac{{| y |}^{9}}{x^{6}} x^{6} = e^{C} x^{6}$

Step 5.5.3.1.1.2

Rewrite the expression.

${| y |}^{9} = e^{C} x^{6}$

Step 5.5.3.2

Simplify the right side.

Tap for more steps...

Step 5.5.3.2.1

Reorder factors in $e^{C} x^{6}$ .

${| y |}^{9} = x^{6} e^{C}$

Step 5.5.4

Solve for $y$ .

Tap for more steps...

Step 5.5.4.1

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

$| y | = \sqrt[9]{x^{6} e^{C}}$

Step 5.5.4.2

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

$y = \pm \sqrt[9]{x^{6} e^{C}}$

Step 6

Simplify the constant of integration.

$y = \pm \sqrt[9]{x^{6} C}$