Calculus Examples

$\frac{d y}{d x} + 7 x = 0$

Step 1

Separate the variables.

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Step 1.1

Subtract $7 x$ from both sides of the equation.

$\frac{d y}{d x} = - 7 x$

Step 1.2

Rewrite the equation.

$d y = - 7 x d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int - 7 x d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int - 7 x d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $- 7$ is constant with respect to $x$ , move $- 7$ out of the integral.

$y + C_{1} = - 7 \int x d x$

Step 2.3.2

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

$y + C_{1} = - 7 (\frac{1}{2} x^{2} + C_{2})$

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite $- 7 (\frac{1}{2} x^{2} + C_{2})$ as $- 7 (\frac{1}{2}) x^{2} + C_{2}$ .

$y + C_{1} = - 7 (\frac{1}{2}) x^{2} + C_{2}$

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

Combine $- 7$ and $\frac{1}{2}$ .

$y + C_{1} = \frac{- 7}{2} x^{2} + C_{2}$

Step 2.3.3.2.2

Move the negative in front of the fraction.

$y + C_{1} = - \frac{7}{2} x^{2} + C_{2}$

Step 2.4

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

$y = - \frac{7}{2} x^{2} + K$