Calculus Examples

$x (\frac{d y}{d x}) = y + 2 x e^{- \frac{y}{x}}$

Step 1

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

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Step 1.1

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

$\frac{x \frac{d y}{d x}}{x} = \frac{y}{x} + \frac{2 x e^{- \frac{y}{x}}}{x}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $x$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{x \frac{d y}{d x}}{x} = \frac{y}{x} + \frac{2 x e^{- \frac{y}{x}}}{x}$

Step 1.2.1.2

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

$\frac{d y}{d x} = \frac{y}{x} + \frac{2 x e^{- \frac{y}{x}}}{x}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Cancel the common factor of $x$ .

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Step 1.3.1.1

Cancel the common factor.

$\frac{d y}{d x} = \frac{y}{x} + \frac{2 x e^{- \frac{y}{x}}}{x}$

Step 1.3.1.2

Divide $2 e^{- \frac{y}{x}}$ by $1$ .

$\frac{d y}{d x} = \frac{y}{x} + 2 e^{- \frac{y}{x}}$

Step 2

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

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

Step 3

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

$y = V x$

Step 4

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 5

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

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

Step 6

Solve the substituted differential equation.

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Step 6.1

Separate the variables.

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Step 6.1.1

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

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Step 6.1.1.1

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

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Step 6.1.1.1.1

Subtract $V$ from both sides of the equation.

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

Step 6.1.1.1.2

Combine the opposite terms in $V + 2 e^{- V} - V$ .

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Step 6.1.1.1.2.1

Subtract $V$ from $V$ .

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

Step 6.1.1.1.2.2

Add $2 e^{- V}$ and $0$ .

$x \frac{d V}{d x} = 2 e^{- V}$

Step 6.1.1.2

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

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Step 6.1.1.2.1

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

$\frac{x \frac{d V}{d x}}{x} = \frac{2 e^{- V}}{x}$

Step 6.1.1.2.2

Simplify the left side.

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Step 6.1.1.2.2.1

Cancel the common factor of $x$ .

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Step 6.1.1.2.2.1.1

Cancel the common factor.

$\frac{x \frac{d V}{d x}}{x} = \frac{2 e^{- V}}{x}$

Step 6.1.1.2.2.1.2

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

$\frac{d V}{d x} = \frac{2 e^{- V}}{x}$

Step 6.1.2

Multiply both sides by $\frac{1}{e^{- V}}$ .

$\frac{1}{e^{- V}} \frac{d V}{d x} = \frac{1}{e^{- V}} \cdot \frac{2 e^{- V}}{x}$

Step 6.1.3

Simplify.

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Step 6.1.3.1

Combine.

$\frac{1}{e^{- V}} \frac{d V}{d x} = \frac{1 (2 e^{- V})}{e^{- V} x}$

Step 6.1.3.2

Cancel the common factor of $e^{- V}$ .

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Step 6.1.3.2.1

Cancel the common factor.

$\frac{1}{e^{- V}} \frac{d V}{d x} = \frac{1 (2 e^{- V})}{e^{- V} x}$

Step 6.1.3.2.2

Rewrite the expression.

$\frac{1}{e^{- V}} \frac{d V}{d x} = \frac{1 \cdot (2)}{x}$

Step 6.1.3.3

Multiply $2$ by $1$ .

$\frac{1}{e^{- V}} \frac{d V}{d x} = \frac{2}{x}$

Step 6.1.4

Rewrite the equation.

$\frac{1}{e^{- V}} d V = \frac{2}{x} d x$

Step 6.2

Integrate both sides.

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Step 6.2.1

Set up an integral on each side.

$\int \frac{1}{e^{- V}} d V = \int \frac{2}{x} d x$

Step 6.2.2

Integrate the left side.

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Step 6.2.2.1

Simplify the expression.

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Step 6.2.2.1.1

Negate the exponent of $e^{- V}$ and move it out of the denominator.

$\int 1 {(e^{- V})}^{- 1} d V = \int \frac{2}{x} d x$

Step 6.2.2.1.2

Simplify.

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Step 6.2.2.1.2.1

Multiply the exponents in ${(e^{- V})}^{- 1}$ .

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Step 6.2.2.1.2.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int 1 e^{- V \cdot - 1} d V = \int \frac{2}{x} d x$

Step 6.2.2.1.2.1.2

Multiply $- V \cdot - 1$ .

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Step 6.2.2.1.2.1.2.1

Multiply $- 1$ by $- 1$ .

$\int 1 e^{1 V} d V = \int \frac{2}{x} d x$

Step 6.2.2.1.2.1.2.2

Multiply $V$ by $1$ .

$\int 1 e^{V} d V = \int \frac{2}{x} d x$

Step 6.2.2.1.2.2

Multiply $e^{V}$ by $1$ .

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

Step 6.2.2.2

The integral of $e^{V}$ with respect to $V$ is $e^{V}$ .

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

Step 6.2.3

Integrate the right side.

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Step 6.2.3.1

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

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

Step 6.2.3.2

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

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

Step 6.2.3.3

Simplify.

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

Step 6.2.4

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

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

Step 6.3

Solve for $V$ .

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Step 6.3.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

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

Step 6.3.2

Expand the left side.

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Step 6.3.2.1

Expand $\ln (e^{V})$ by moving $V$ outside the logarithm.

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

Step 6.3.2.2

The natural logarithm of $e$ is $1$ .

$V \cdot 1 = \ln (\ln (x^{2}) + C)$

Step 6.3.2.3

Multiply $V$ by $1$ .

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

Step 6.3.3

Expand $\ln (x^{2})$ by moving $2$ outside the logarithm.

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

Step 6.3.4

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

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

Step 7

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

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

Step 8

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

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Step 8.1

Multiply both sides by $x$ .

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

Step 8.2

Simplify.

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Step 8.2.1

Simplify the left side.

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Step 8.2.1.1

Cancel the common factor of $x$ .

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Step 8.2.1.1.1

Cancel the common factor.

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

Step 8.2.1.1.2

Rewrite the expression.

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

Step 8.2.2

Simplify the right side.

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Step 8.2.2.1

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

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