Calculus Examples

$y d y = x y d x$

Step 1

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

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

Step 2

Simplify.

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Step 2.1

Cancel the common factor of $y$ .

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Step 2.1.1

Cancel the common factor.

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

Step 2.1.2

Rewrite the expression.

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

Step 2.2

Cancel the common factor of $y$ .

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Step 2.2.1

Factor $y$ out of $x y$ .

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

Step 2.2.2

Cancel the common factor.

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

Step 2.2.3

Rewrite the expression.

$1 d y = x d x$

Step 3

Integrate both sides.

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Step 3.1

Set up an integral on each side.

$\int d y = \int x d x$

Step 3.2

Apply the constant rule.

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

Step 3.3

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

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

Step 3.4

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

$y = \frac{1}{2} x^{2} + K$