Calculus Examples

$5 \frac{d y}{d x} + 2 x = 3$

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

Subtract $2 x$ from both sides of the equation.

$5 \frac{d y}{d x} = 3 - 2 x$

Step 1.1.2

Divide each term in $5 \frac{d y}{d x} = 3 - 2 x$ by $5$ and simplify.

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Step 1.1.2.1

Divide each term in $5 \frac{d y}{d x} = 3 - 2 x$ by $5$ .

$\frac{5 \frac{d y}{d x}}{5} = \frac{3}{5} + \frac{- 2 x}{5}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $5$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{5 \frac{d y}{d x}}{5} = \frac{3}{5} + \frac{- 2 x}{5}$

Step 1.1.2.2.1.2

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

$\frac{d y}{d x} = \frac{3}{5} + \frac{- 2 x}{5}$

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

Move the negative in front of the fraction.

$\frac{d y}{d x} = \frac{3}{5} - \frac{2 x}{5}$

Step 1.2

Rewrite the equation.

$d y = (\frac{3}{5} - \frac{2 x}{5}) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int \frac{3}{5} - \frac{2 x}{5} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int \frac{3}{5} - \frac{2 x}{5} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$y + C_{1} = \int \frac{3}{5} d x + \int - \frac{2 x}{5} d x$

Step 2.3.2

Apply the constant rule.

$y + C_{1} = \frac{3}{5} x + C_{2} + \int - \frac{2 x}{5} d x$

Step 2.3.3

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$y + C_{1} = \frac{3}{5} x + C_{2} - \int \frac{2 x}{5} d x$

Step 2.3.4

Since $\frac{2}{5}$ is constant with respect to $x$ , move $\frac{2}{5}$ out of the integral.

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

Step 2.3.5

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

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

Step 2.3.6

Simplify.

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Step 2.3.6.1

Simplify.

$y + C_{1} = \frac{3}{5} x - \frac{2}{5} \cdot \frac{1}{2} x^{2} + C_{4}$

Step 2.3.6.2

Simplify.

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Step 2.3.6.2.1

Multiply $\frac{1}{2}$ by $\frac{2}{5}$ .

$y + C_{1} = \frac{3}{5} x - \frac{2}{2 \cdot 5} x^{2} + C_{4}$

Step 2.3.6.2.2

Multiply $2$ by $5$ .

$y + C_{1} = \frac{3}{5} x - \frac{2}{10} x^{2} + C_{4}$

Step 2.3.6.2.3

Cancel the common factor of $2$ and $10$ .

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Step 2.3.6.2.3.1

Factor $2$ out of $2$ .

$y + C_{1} = \frac{3}{5} x - \frac{2 (1)}{10} x^{2} + C_{4}$

Step 2.3.6.2.3.2

Cancel the common factors.

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Step 2.3.6.2.3.2.1

Factor $2$ out of $10$ .

$y + C_{1} = \frac{3}{5} x - \frac{2 \cdot 1}{2 \cdot 5} x^{2} + C_{4}$

Step 2.3.6.2.3.2.2

Cancel the common factor.

$y + C_{1} = \frac{3}{5} x - \frac{2 \cdot 1}{2 \cdot 5} x^{2} + C_{4}$

Step 2.3.6.2.3.2.3

Rewrite the expression.

$y + C_{1} = \frac{3}{5} x - \frac{1}{5} x^{2} + C_{4}$

Step 2.4

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

$y = \frac{3}{5} x - \frac{1}{5} x^{2} + K$