Calculus Examples

$2 \frac{d y}{d x} - x = 10$

Step 1

Separate the variables.

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Step 1.1

Solve for $\frac{d y}{d x}$ .

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Step 1.1.1

Add $x$ to both sides of the equation.

$2 \frac{d y}{d x} = 10 + x$

Step 1.1.2

Divide each term in $2 \frac{d y}{d x} = 10 + x$ by $2$ and simplify.

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Step 1.1.2.1

Divide each term in $2 \frac{d y}{d x} = 10 + x$ by $2$ .

$\frac{2 \frac{d y}{d x}}{2} = \frac{10}{2} + \frac{x}{2}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $2$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{2 \frac{d y}{d x}}{2} = \frac{10}{2} + \frac{x}{2}$

Step 1.1.2.2.1.2

Divide $\frac{d y}{d x}$ by $1$ .

$\frac{d y}{d x} = \frac{10}{2} + \frac{x}{2}$

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

Divide $10$ by $2$ .

$\frac{d y}{d x} = 5 + \frac{x}{2}$

Step 1.2

Rewrite the equation.

$d y = (5 + \frac{x}{2}) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int 5 + \frac{x}{2} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 5 + \frac{x}{2} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$y + C_{1} = \int 5 d x + \int \frac{x}{2} d x$

Step 2.3.2

Apply the constant rule.

$y + C_{1} = 5 x + C_{2} + \int \frac{x}{2} d x$

Step 2.3.3

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

$y + C_{1} = 5 x + C_{2} + \frac{1}{2} \int x d x$

Step 2.3.4

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

$y + C_{1} = 5 x + C_{2} + \frac{1}{2} (\frac{1}{2} x^{2} + C_{3})$

Step 2.3.5

Simplify.

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Step 2.3.5.1

Simplify.

$y + C_{1} = 5 x + \frac{1}{2} \cdot \frac{1}{2} x^{2} + C_{4}$

Step 2.3.5.2

Simplify.

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Step 2.3.5.2.1

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

$y + C_{1} = 5 x + \frac{1}{2 \cdot 2} x^{2} + C_{4}$

Step 2.3.5.2.2

Multiply $2$ by $2$ .

$y + C_{1} = 5 x + \frac{1}{4} x^{2} + C_{4}$

Step 2.3.6

Reorder terms.

$y + C_{1} = \frac{1}{4} x^{2} + 5 x + C_{4}$

Step 2.4

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

$y = \frac{1}{4} x^{2} + 5 x + K$