Calculus Examples

$3 x d y - 4 y d x = 0$

Step 1

Add $4 y d x$ to both sides of the equation.

$3 x d y = 4 y d x$

Step 2

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

$\frac{1}{x y} (3 x) d y = \frac{1}{x y} (4 y) d x$

Step 3

Simplify.

Tap for more steps...

Step 3.1

Rewrite using the commutative property of multiplication.

$3 \frac{1}{x y} x d y = \frac{1}{x y} (4 y) d x$

Step 3.2

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

$\frac{3}{x y} x d y = \frac{1}{x y} (4 y) d x$

Step 3.3

Cancel the common factor of $x$ .

Tap for more steps...

Step 3.3.1

Factor $x$ out of $x y$ .

$\frac{3}{x (y)} x d y = \frac{1}{x y} (4 y) d x$

Step 3.3.2

Cancel the common factor.

$\frac{3}{x y} x d y = \frac{1}{x y} (4 y) d x$

Step 3.3.3

Rewrite the expression.

$\frac{3}{y} d y = \frac{1}{x y} (4 y) d x$

Step 3.4

Rewrite using the commutative property of multiplication.

$\frac{3}{y} d y = 4 \frac{1}{x y} y d x$

Step 3.5

Combine $4$ and $\frac{1}{x y}$ .

$\frac{3}{y} d y = \frac{4}{x y} y d x$

Step 3.6

Cancel the common factor of $y$ .

Tap for more steps...

Step 3.6.1

Factor $y$ out of $x y$ .

$\frac{3}{y} d y = \frac{4}{y x} y d x$

Step 3.6.2

Cancel the common factor.

$\frac{3}{y} d y = \frac{4}{y x} y d x$

Step 3.6.3

Rewrite the expression.

$\frac{3}{y} d y = \frac{4}{x} d x$

Step 4

Integrate both sides.

Tap for more steps...

Step 4.1

Set up an integral on each side.

$\int \frac{3}{y} d y = \int \frac{4}{x} d x$

Step 4.2

Integrate the left side.

Tap for more steps...

Step 4.2.1

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

$3 \int \frac{1}{y} d y = \int \frac{4}{x} d x$

Step 4.2.2

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

$3 (\ln (| y |) + C_{1}) = \int \frac{4}{x} d x$

Step 4.2.3

Simplify.

$3 \ln (| y |) + C_{1} = \int \frac{4}{x} d x$

Step 4.3

Integrate the right side.

Tap for more steps...

Step 4.3.1

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$3 \ln (| y |) + C_{1} = 4 \int \frac{1}{x} d x$

Step 4.3.2

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

$3 \ln (| y |) + C_{1} = 4 (\ln (| x |) + C_{2})$

Step 4.3.3

Simplify.

$3 \ln (| y |) + C_{1} = 4 \ln (| x |) + C_{2}$

Step 4.4

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

$3 \ln (| y |) = 4 \ln (| x |) + C$

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.

$3 \ln (| y |) - 4 \ln (| x |) = C$

Step 5.2

Simplify the left side.

Tap for more steps...

Step 5.2.1

Simplify $3 \ln (| y |) - 4 \ln (| x |)$ .

Tap for more steps...

Step 5.2.1.1

Simplify each term.

Tap for more steps...

Step 5.2.1.1.1

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

$\ln ({| y |}^{3}) - 4 \ln (| x |) = C$

Step 5.2.1.1.2

Simplify $- 4 \ln (| x |)$ by moving $4$ inside the logarithm.

$\ln ({| y |}^{3}) - \ln ({| x |}^{4}) = C$

Step 5.2.1.1.3

Remove the absolute value in ${| x |}^{4}$ because exponentiations with even powers are always positive.

$\ln ({| y |}^{3}) - \ln (x^{4}) = C$

Step 5.2.1.2

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

$\ln (\frac{{| y |}^{3}}{x^{4}}) = C$

Step 5.3

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

$e^{\ln (\frac{{| y |}^{3}}{x^{4}})} = e^{C}$

Step 5.4

Rewrite $\ln (\frac{{| y |}^{3}}{x^{4}}) = C$ 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 $b^{y} = x$ .

$e^{C} = \frac{{| y |}^{3}}{x^{4}}$

Step 5.5

Solve for $y$ .

Tap for more steps...

Step 5.5.1

Rewrite the equation as $\frac{{| y |}^{3}}{x^{4}} = e^{C}$ .

$\frac{{| y |}^{3}}{x^{4}} = e^{C}$

Step 5.5.2

Multiply both sides by $x^{4}$ .

$\frac{{| y |}^{3}}{x^{4}} x^{4} = e^{C} x^{4}$

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 $x^{4}$ .

Tap for more steps...

Step 5.5.3.1.1.1

Cancel the common factor.

$\frac{{| y |}^{3}}{x^{4}} x^{4} = e^{C} x^{4}$

Step 5.5.3.1.1.2

Rewrite the expression.

${| y |}^{3} = e^{C} x^{4}$

Step 5.5.3.2

Simplify the right side.

Tap for more steps...

Step 5.5.3.2.1

Reorder factors in $e^{C} x^{4}$ .

${| y |}^{3} = x^{4} e^{C}$

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.

$| y | = \sqrt[3]{x^{4} e^{C}}$

Step 5.5.4.2

Simplify $\sqrt[3]{x^{4} e^{C}}$ .

Tap for more steps...

Step 5.5.4.2.1

Rewrite $x^{4} e^{C}$ as $x^{3} (x e^{C})$ .

Tap for more steps...

Step 5.5.4.2.1.1

Factor out $x^{3}$ .

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

Step 5.5.4.2.1.2

Add parentheses.

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

Step 5.5.4.2.2

Pull terms out from under the radical.

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

Step 5.5.4.3

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

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

Step 6

Simplify the constant of integration.

$y = \pm x \sqrt[3]{x C}$