Calculus Examples

$x^{2} y d y - y d x = 0$

Step 1

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

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

Step 2

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

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

Step 3

Simplify.

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Step 3.1

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

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $y$ .

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

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

Step 4.2

Apply the constant rule.

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

Step 4.3

Integrate the right side.

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Step 4.3.1

Apply basic rules of exponents.

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Step 4.3.1.1

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

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

Step 4.3.1.2

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

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Step 4.3.1.2.1

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

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

Step 4.3.1.2.2

Multiply $2$ by $- 1$ .

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

Step 4.3.2

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

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

Step 4.3.3

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

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

Step 4.4

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

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