Calculus Examples

$\frac{d y}{d x} - X Y = 5 X$

Step 1

Separate the variables.

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Step 1.1

Add $X Y$ to both sides of the equation.

$\frac{d y}{d x} = 5 X + X Y$

Step 1.2

Rewrite the equation.

$d y = (5 X + X Y) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int 5 X + X Y d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 5 X + X Y d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Apply the constant rule.

$y + C_{1} = (5 X + X Y) x + C_{2}$

Step 2.3.2

Reorder terms.

$y + C_{1} = x (5 X + X Y) + C_{2}$

Step 2.4

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

$y = x (5 X + X Y) + K$