Calculus Examples

$\frac{d y}{d x} + 3 x^{2} y^{2} = 4 x^{3} y^{2}$

Step 1

Separate the variables.

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Step 1.1

Subtract $3 x^{2} y^{2}$ from both sides of the equation.

$\frac{d y}{d x} = 4 x^{3} y^{2} - 3 x^{2} y^{2}$

Step 1.2

Factor $x^{2} y^{2}$ out of $4 x^{3} y^{2} - 3 x^{2} y^{2}$ .

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Step 1.2.1

Factor $x^{2} y^{2}$ out of $4 x^{3} y^{2}$ .

$\frac{d y}{d x} = x^{2} y^{2} (4 x) - 3 x^{2} y^{2}$

Step 1.2.2

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

$\frac{d y}{d x} = x^{2} y^{2} (4 x) + x^{2} y^{2} (- 3)$

Step 1.2.3

Factor $x^{2} y^{2}$ out of $x^{2} y^{2} (4 x) + x^{2} y^{2} (- 3)$ .

$\frac{d y}{d x} = x^{2} y^{2} (4 x - 3)$

Step 1.3

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

$\frac{1}{y^{2}} \frac{d y}{d x} = \frac{1}{y^{2}} (x^{2} y^{2} (4 x - 3))$

Step 1.4

Simplify.

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Step 1.4.1

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

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Step 1.4.1.1

Factor $y^{2}$ out of $x^{2} y^{2} (4 x - 3)$ .

$\frac{1}{y^{2}} \frac{d y}{d x} = \frac{1}{y^{2}} (y^{2} (x^{2} (4 x - 3)))$

Step 1.4.1.2

Cancel the common factor.

$\frac{1}{y^{2}} \frac{d y}{d x} = \frac{1}{y^{2}} (y^{2} (x^{2} (4 x - 3)))$

Step 1.4.1.3

Rewrite the expression.

$\frac{1}{y^{2}} \frac{d y}{d x} = x^{2} (4 x - 3)$

Step 1.4.2

Apply the distributive property.

$\frac{1}{y^{2}} \frac{d y}{d x} = x^{2} (4 x) + x^{2} \cdot - 3$

Step 1.4.3

Rewrite using the commutative property of multiplication.

$\frac{1}{y^{2}} \frac{d y}{d x} = 4 x^{2} x + x^{2} \cdot - 3$

Step 1.4.4

Move $- 3$ to the left of $x^{2}$ .

$\frac{1}{y^{2}} \frac{d y}{d x} = 4 x^{2} x - 3 \cdot x^{2}$

Step 1.4.5

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.4.5.1

Move $x$ .

$\frac{1}{y^{2}} \frac{d y}{d x} = 4 (x \cdot x^{2}) - 3 \cdot x^{2}$

Step 1.4.5.2

Multiply $x$ by $x^{2}$ .

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Step 1.4.5.2.1

Raise $x$ to the power of $1$ .

$\frac{1}{y^{2}} \frac{d y}{d x} = 4 (x^{1} x^{2}) - 3 \cdot x^{2}$

Step 1.4.5.2.2

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

$\frac{1}{y^{2}} \frac{d y}{d x} = 4 x^{1 + 2} - 3 \cdot x^{2}$

Step 1.4.5.3

Add $1$ and $2$ .

$\frac{1}{y^{2}} \frac{d y}{d x} = 4 x^{3} - 3 \cdot x^{2}$

$\frac{1}{y^{2}} \frac{d y}{d x} = 4 x^{3} - 3 x^{2}$

Step 1.5

Rewrite the equation.

$\frac{1}{y^{2}} d y = (4 x^{3} - 3 x^{2}) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{y^{2}} d y = \int 4 x^{3} - 3 x^{2} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Apply basic rules of exponents.

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Step 2.2.1.1

Move $y^{2}$ out of the denominator by raising it to the $- 1$ power.

$\int {(y^{2})}^{- 1} d y = \int 4 x^{3} - 3 x^{2} d x$

Step 2.2.1.2

Multiply the exponents in ${(y^{2})}^{- 1}$ .

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Step 2.2.1.2.1

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

$\int y^{2 \cdot - 1} d y = \int 4 x^{3} - 3 x^{2} d x$

Step 2.2.1.2.2

Multiply $2$ by $- 1$ .

$\int y^{- 2} d y = \int 4 x^{3} - 3 x^{2} d x$

Step 2.2.2

By the Power Rule, the integral of $y^{- 2}$ with respect to $y$ is $- y^{- 1}$ .

$- y^{- 1} + C_{1} = \int 4 x^{3} - 3 x^{2} d x$

Step 2.2.3

Rewrite $- y^{- 1} + C_{1}$ as $- \frac{1}{y} + C_{1}$ .

$- \frac{1}{y} + C_{1} = \int 4 x^{3} - 3 x^{2} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$- \frac{1}{y} + C_{1} = \int 4 x^{3} d x + \int - 3 x^{2} d x$

Step 2.3.2

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

$- \frac{1}{y} + C_{1} = 4 \int x^{3} d x + \int - 3 x^{2} d x$

Step 2.3.3

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$- \frac{1}{y} + C_{1} = 4 (\frac{1}{4} x^{4} + C_{2}) + \int - 3 x^{2} d x$

Step 2.3.4

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

$- \frac{1}{y} + C_{1} = 4 (\frac{1}{4} x^{4} + C_{2}) - 3 \int x^{2} d x$

Step 2.3.5

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$- \frac{1}{y} + C_{1} = 4 (\frac{1}{4} x^{4} + C_{2}) - 3 (\frac{1}{3} x^{3} + C_{3})$

Step 2.3.6

Simplify.

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Step 2.3.6.1

Simplify.

$- \frac{1}{y} + C_{1} = 4 (\frac{1}{4}) x^{4} - 3 (\frac{1}{3}) x^{3} + C_{4}$

Step 2.3.6.2

Simplify.

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Step 2.3.6.2.1

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

$- \frac{1}{y} + C_{1} = \frac{4}{4} x^{4} - 3 (\frac{1}{3}) x^{3} + C_{4}$

Step 2.3.6.2.2

Cancel the common factor of $4$ .

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Step 2.3.6.2.2.1

Cancel the common factor.

$- \frac{1}{y} + C_{1} = \frac{4}{4} x^{4} - 3 (\frac{1}{3}) x^{3} + C_{4}$

Step 2.3.6.2.2.2

Rewrite the expression.

$- \frac{1}{y} + C_{1} = 1 x^{4} - 3 (\frac{1}{3}) x^{3} + C_{4}$

Step 2.3.6.2.3

Multiply $x^{4}$ by $1$ .

$- \frac{1}{y} + C_{1} = x^{4} - 3 (\frac{1}{3}) x^{3} + C_{4}$

Step 2.3.6.2.4

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

$- \frac{1}{y} + C_{1} = x^{4} + \frac{- 3}{3} x^{3} + C_{4}$

Step 2.3.6.2.5

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

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Step 2.3.6.2.5.1

Factor $3$ out of $- 3$ .

$- \frac{1}{y} + C_{1} = x^{4} + \frac{3 \cdot - 1}{3} x^{3} + C_{4}$

Step 2.3.6.2.5.2

Cancel the common factors.

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Step 2.3.6.2.5.2.1

Factor $3$ out of $3$ .

$- \frac{1}{y} + C_{1} = x^{4} + \frac{3 \cdot - 1}{3 (1)} x^{3} + C_{4}$

Step 2.3.6.2.5.2.2

Cancel the common factor.

$- \frac{1}{y} + C_{1} = x^{4} + \frac{3 \cdot - 1}{3 \cdot 1} x^{3} + C_{4}$

Step 2.3.6.2.5.2.3

Rewrite the expression.

$- \frac{1}{y} + C_{1} = x^{4} + \frac{- 1}{1} x^{3} + C_{4}$

Step 2.3.6.2.5.2.4

Divide $- 1$ by $1$ .

$- \frac{1}{y} + C_{1} = x^{4} - x^{3} + C_{4}$

Step 2.4

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

$- \frac{1}{y} = x^{4} - x^{3} + K$

Step 3

Solve for $y$ .

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Step 3.1

Find the LCD of the terms in the equation.

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Step 3.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y, 1, 1, 1$

Step 3.1.2

The LCM of one and any expression is the expression.

$y$

Step 3.2

Multiply each term in $- \frac{1}{y} = x^{4} - x^{3} + K$ by $y$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $- \frac{1}{y} = x^{4} - x^{3} + K$ by $y$ .

$- \frac{1}{y} y = x^{4} y - x^{3} y + K y$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $y$ .

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Step 3.2.2.1.1

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

$\frac{- 1}{y} y = x^{4} y - x^{3} y + K y$

Step 3.2.2.1.2

Cancel the common factor.

$\frac{- 1}{y} y = x^{4} y - x^{3} y + K y$

Step 3.2.2.1.3

Rewrite the expression.

$- 1 = x^{4} y - x^{3} y + K y$

Step 3.3

Solve the equation.

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Step 3.3.1

Rewrite the equation as $x^{4} y - x^{3} y + K y = - 1$ .

$x^{4} y - x^{3} y + K y = - 1$

Step 3.3.2

Factor $y$ out of $x^{4} y - x^{3} y + K y$ .

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Step 3.3.2.1

Factor $y$ out of $x^{4} y$ .

$y x^{4} - x^{3} y + K y = - 1$

Step 3.3.2.2

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

$y x^{4} + y (- x^{3}) + K y = - 1$

Step 3.3.2.3

Factor $y$ out of $K y$ .

$y x^{4} + y (- x^{3}) + y K = - 1$

Step 3.3.2.4

Factor $y$ out of $y x^{4} + y (- x^{3})$ .

$y (x^{4} - x^{3}) + y K = - 1$

Step 3.3.2.5

Factor $y$ out of $y (x^{4} - x^{3}) + y K$ .

$y (x^{4} - x^{3} + K) = - 1$

Step 3.3.3

Divide each term in $y (x^{4} - x^{3} + K) = - 1$ by $x^{4} - x^{3} + K$ and simplify.

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Step 3.3.3.1

Divide each term in $y (x^{4} - x^{3} + K) = - 1$ by $x^{4} - x^{3} + K$ .

$\frac{y (x^{4} - x^{3} + K)}{x^{4} - x^{3} + K} = \frac{- 1}{x^{4} - x^{3} + K}$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $x^{4} - x^{3} + K$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{y (x^{4} - x^{3} + K)}{x^{4} - x^{3} + K} = \frac{- 1}{x^{4} - x^{3} + K}$

Step 3.3.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 1}{x^{4} - x^{3} + K}$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Move the negative in front of the fraction.

$y = - \frac{1}{x^{4} - x^{3} + K}$