Calculus Examples

$- 3 (y d) x + 2 x d y = 0$

Step 1

Add $3 y d x$ to both sides of the equation.

$2 x d y = 3 y d x$

Step 2

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

$\frac{1}{x y} (2 x) d y = \frac{1}{x y} (3 y) d x$

Step 3

Simplify.

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Step 3.1

Rewrite using the commutative property of multiplication.

$2 \frac{1}{x y} x d y = \frac{1}{x y} (3 y) d x$

Step 3.2

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

$\frac{2}{x y} x d y = \frac{1}{x y} (3 y) d x$

Step 3.3

Cancel the common factor of $x$ .

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Step 3.3.1

Factor $x$ out of $x y$ .

$\frac{2}{x (y)} x d y = \frac{1}{x y} (3 y) d x$

Step 3.3.2

Cancel the common factor.

$\frac{2}{x y} x d y = \frac{1}{x y} (3 y) d x$

Step 3.3.3

Rewrite the expression.

$\frac{2}{y} d y = \frac{1}{x y} (3 y) d x$

Step 3.4

Rewrite using the commutative property of multiplication.

$\frac{2}{y} d y = 3 \frac{1}{x y} y d x$

Step 3.5

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

$\frac{2}{y} d y = \frac{3}{x y} y d x$

Step 3.6

Cancel the common factor of $y$ .

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Step 3.6.1

Factor $y$ out of $x y$ .

$\frac{2}{y} d y = \frac{3}{y x} y d x$

Step 3.6.2

Cancel the common factor.

$\frac{2}{y} d y = \frac{3}{y x} y d x$

Step 3.6.3

Rewrite the expression.

$\frac{2}{y} d y = \frac{3}{x} d x$

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

$\int \frac{2}{y} d y = \int \frac{3}{x} d x$

Step 4.2

Integrate the left side.

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Step 4.2.1

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

$2 \int \frac{1}{y} d y = \int \frac{3}{x} d x$

Step 4.2.2

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

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

Step 4.2.3

Simplify.

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

Step 4.3

Integrate the right side.

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Step 4.3.1

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

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

Step 4.3.2

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

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

Step 4.3.3

Simplify.

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

Step 4.4

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

$2 \ln (| y |) = 3 \ln (| x |) + C$

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.

$2 \ln (| y |) - 3 \ln (| x |) = C$

Step 5.2

Simplify the left side.

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Step 5.2.1

Simplify $2 \ln (| y |) - 3 \ln (| x |)$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

Simplify $2 \ln (| y |)$ by moving $2$ inside the logarithm.

$\ln ({| y |}^{2}) - 3 \ln (| x |) = C$

Step 5.2.1.1.2

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

$\ln (y^{2}) - 3 \ln (| x |) = C$

Step 5.2.1.1.3

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

$\ln (y^{2}) - \ln ({| x |}^{3}) = C$

Step 5.2.1.2

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

$\ln (\frac{y^{2}}{{| x |}^{3}}) = C$

Step 5.3

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

$e^{\ln (\frac{y^{2}}{{| x |}^{3}})} = e^{C}$

Step 5.4

Rewrite $\ln (\frac{y^{2}}{{| x |}^{3}}) = C$ 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 $b^{y} = x$ .

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

Step 5.5

Solve for $y$ .

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Step 5.5.1

Rewrite the equation as $\frac{y^{2}}{{| x |}^{3}} = e^{C}$ .

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

Step 5.5.2

Multiply both sides by ${| x |}^{3}$ .

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

Step 5.5.3

Simplify the left side.

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Step 5.5.3.1

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

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Step 5.5.3.1.1

Cancel the common factor.

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

Step 5.5.3.1.2

Rewrite the expression.

$y^{2} = e^{C} {| x |}^{3}$

Step 5.5.4

Solve for $y$ .

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Step 5.5.4.1

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

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

Step 5.5.4.2

Simplify $\pm \sqrt{e^{C} {| x |}^{3}}$ .

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Step 5.5.4.2.1

Rewrite $e^{C} {| x |}^{3}$ as ${| x |}^{2} (e^{C} | x |)$ .

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Step 5.5.4.2.1.1

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

$y = \pm \sqrt{e^{C} ({| x |}^{2} | x |)}$

Step 5.5.4.2.1.2

Reorder $e^{C}$ and ${| x |}^{2}$ .

$y = \pm \sqrt{{| x |}^{2} e^{C} | x |}$

Step 5.5.4.2.1.3

Add parentheses.

$y = \pm \sqrt{{| x |}^{2} (e^{C} | x |)}$

Step 5.5.4.2.2

Pull terms out from under the radical.

$y = \pm | x | \sqrt{e^{C} | x |}$

Step 5.5.4.3

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

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Step 5.5.4.3.1

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