Calculus Examples

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

Step 1

Let $V = \frac{y}{x}$ . Substitute $V$ for $\frac{y}{x}$ .

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

Step 2

Solve $V = \frac{y}{x}$ for $y$ .

$y = V x$

Step 3

Use the product rule to find the derivative of $y = V x$ with respect to $x$ .

$\frac{d y}{d x} = x \frac{d V}{d x} + V$

Step 4

Substitute $x \frac{d V}{d x} + V$ for $\frac{d y}{d x}$ .

$x \frac{d V}{d x} + V = V + 2$

Step 5

Solve the substituted differential equation.

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Step 5.1

Separate the variables.

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Step 5.1.1

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

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Step 5.1.1.1

Move all terms not containing $\frac{d V}{d x}$ to the right side of the equation.

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Step 5.1.1.1.1

Subtract $V$ from both sides of the equation.

$x \frac{d V}{d x} = V + 2 - V$

Step 5.1.1.1.2

Combine the opposite terms in $V + 2 - V$ .

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Step 5.1.1.1.2.1

Subtract $V$ from $V$ .

$x \frac{d V}{d x} = 0 + 2$

Step 5.1.1.1.2.2

Add $0$ and $2$ .

$x \frac{d V}{d x} = 2$

Step 5.1.1.2

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

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Step 5.1.1.2.1

Divide each term in $x \frac{d V}{d x} = 2$ by $x$ .

$\frac{x \frac{d V}{d x}}{x} = \frac{2}{x}$

Step 5.1.1.2.2

Simplify the left side.

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Step 5.1.1.2.2.1

Cancel the common factor of $x$ .

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Step 5.1.1.2.2.1.1

Cancel the common factor.

$\frac{x \frac{d V}{d x}}{x} = \frac{2}{x}$

Step 5.1.1.2.2.1.2

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

$\frac{d V}{d x} = \frac{2}{x}$

Step 5.1.2

Rewrite the equation.

$d V = \frac{2}{x} d x$

Step 5.2

Integrate both sides.

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Step 5.2.1

Set up an integral on each side.

$\int d V = \int \frac{2}{x} d x$

Step 5.2.2

Apply the constant rule.

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

Step 5.2.3

Integrate the right side.

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Step 5.2.3.1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 5.2.3.2

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$V + C_{1} = 2 (\ln (| x |) + C_{2})$

Step 5.2.3.3

Simplify.

$V + C_{1} = 2 \ln (| x |) + C_{2}$

Step 5.2.4

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

$V = 2 \ln (| x |) + C$

Step 6

Substitute $\frac{y}{x}$ for $V$ .

$\frac{y}{x} = 2 \ln (| x |) + C$

Step 7

Solve $\frac{y}{x} = 2 \ln (| x |) + C$ for $y$ .

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Step 7.1

Multiply both sides by $x$ .

$\frac{y}{x} x = (2 \ln (| x |) + C) x$

Step 7.2

Simplify.

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Step 7.2.1

Simplify the left side.

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Step 7.2.1.1

Cancel the common factor of $x$ .

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Step 7.2.1.1.1

Cancel the common factor.

$\frac{y}{x} x = (2 \ln (| x |) + C) x$

Step 7.2.1.1.2

Rewrite the expression.

$y = (2 \ln (| x |) + C) x$

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

Simplify $(2 \ln (| x |) + C) x$ .

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Step 7.2.2.1.1

Simplify each term.

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Step 7.2.2.1.1.1

Simplify $2 \ln (| x |)$ by moving $2$ inside the logarithm.

$y = (\ln ({| x |}^{2}) + C) x$

Step 7.2.2.1.1.2

Remove the absolute value in ${| x |}^{2}$ because exponentiations with even powers are always positive.

$y = (\ln (x^{2}) + C) x$

Step 7.2.2.1.2

Simplify by multiplying through.

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Step 7.2.2.1.2.1

Apply the distributive property.

$y = \ln (x^{2}) x + C x$

Step 7.2.2.1.2.2

Simplify the expression.

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Step 7.2.2.1.2.2.1

Reorder factors in $\ln (x^{2}) x + C x$ .

$y = x \ln (x^{2}) + C x$

Step 7.2.2.1.2.2.2

Reorder $x \ln (x^{2})$ and $C x$ .

$y = C x + x \ln (x^{2})$