Calculus Examples

$yd y + 3 x^{2} yd x = 0$

Step 1

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

$y d y = - 3 x^{2} y d x$

Step 2

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

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

Step 3

Simplify.

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Step 3.1

Cancel the common factor of $y$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

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

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

Step 3.4

Cancel the common factor of $y$ .

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Step 3.4.1

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

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

Step 3.4.2

Cancel the common factor.

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

Step 3.4.3

Rewrite the expression.

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

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

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

Step 4.2

Apply the constant rule.

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

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.

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

Step 4.3.2

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

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

Step 4.3.3

Simplify the answer.

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Step 4.3.3.1

Rewrite $- 3 (\frac{1}{3} x^{3} + C_{2})$ as $- 3 (\frac{1}{3}) x^{3} + C_{2}$ .

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

Step 4.3.3.2

Simplify.

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Step 4.3.3.2.1

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

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

Step 4.3.3.2.2

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

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Step 4.3.3.2.2.1

Factor $3$ out of $- 3$ .

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

Step 4.3.3.2.2.2

Cancel the common factors.

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Step 4.3.3.2.2.2.1

Factor $3$ out of $3$ .

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

Step 4.3.3.2.2.2.2

Cancel the common factor.

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

Step 4.3.3.2.2.2.3

Rewrite the expression.

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

Step 4.3.3.2.2.2.4

Divide $- 1$ by $1$ .

$y + C_{1} = - x^{3} + C_{2}$

Step 4.4

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

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