Calculus Examples

$y^{2} \frac{d x}{d y} + x y = 2 y^{2} + 1$

Step 1

Rewrite the differential equation as $\frac{d x}{d y} + M (y) x = Q (y)$ .

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Step 1.1

Factor $x$ out of $x y$ .

$y^{2} \frac{d x}{d y} + x (y) = 2 y^{2} + 1$

Step 1.2

Reorder $x$ and $y$ .

$y^{2} \frac{d x}{d y} + y x = 2 y^{2} + 1$

Step 1.3

Divide each term in $y^{2} \frac{d x}{d y} + y x = 2 y^{2} + 1$ by $y^{2}$ .

$\frac{y^{2} \frac{d x}{d y}}{y^{2}} + \frac{y x}{y^{2}} = \frac{2 y^{2}}{y^{2}} + \frac{1}{y^{2}}$

Step 1.4

Cancel the common factor of $y^{2}$ .

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Step 1.4.1

Cancel the common factor.

$\frac{y^{2} \frac{d x}{d y}}{y^{2}} + \frac{y x}{y^{2}} = \frac{2 y^{2}}{y^{2}} + \frac{1}{y^{2}}$

Step 1.4.2

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

$\frac{d x}{d y} + \frac{y x}{y^{2}} = \frac{2 y^{2}}{y^{2}} + \frac{1}{y^{2}}$

Step 1.5

Cancel the common factor of $y$ and $y^{2}$ .

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Step 1.5.1

Factor $y$ out of $y x$ .

$\frac{d x}{d y} + \frac{y (x)}{y^{2}} = \frac{2 y^{2}}{y^{2}} + \frac{1}{y^{2}}$

Step 1.5.2

Cancel the common factors.

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Step 1.5.2.1

Factor $y$ out of $y^{2}$ .

$\frac{d x}{d y} + \frac{y (x)}{y \cdot y} = \frac{2 y^{2}}{y^{2}} + \frac{1}{y^{2}}$

Step 1.5.2.2

Cancel the common factor.

$\frac{d x}{d y} + \frac{y x}{y \cdot y} = \frac{2 y^{2}}{y^{2}} + \frac{1}{y^{2}}$

Step 1.5.2.3

Rewrite the expression.

$\frac{d x}{d y} + \frac{x}{y} = \frac{2 y^{2}}{y^{2}} + \frac{1}{y^{2}}$

Step 1.6

Cancel the common factor of $y^{2}$ .

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Step 1.6.1

Cancel the common factor.

$\frac{d x}{d y} + \frac{x}{y} = \frac{2 y^{2}}{y^{2}} + \frac{1}{y^{2}}$

Step 1.6.2

Divide $2$ by $1$ .

$\frac{d x}{d y} + \frac{x}{y} = 2 + \frac{1}{y^{2}}$

Step 1.7

Factor $x$ out of $\frac{x}{y}$ .

$\frac{d x}{d y} + x \frac{1}{y} = 2 + \frac{1}{y^{2}}$

Step 1.8

Reorder $x$ and $\frac{1}{y}$ .

$\frac{d x}{d y} + \frac{1}{y} x = 2 + \frac{1}{y^{2}}$

Step 2

The integrating factor is defined by the formula $e^{\int P (y) d y}$ , where $P (y) = \frac{1}{y}$ .

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Step 2.1

Set up the integration.

$e^{\int \frac{1}{y} d y}$

Step 2.2

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

$e^{\ln (| y |) + C}$

Step 2.3

Remove the constant of integration.

$e^{\ln (y)}$

Step 2.4

Exponentiation and log are inverse functions.

$y$

Step 3

Multiply each term by the integrating factor $y$ .

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Step 3.1

Multiply each term by $y$ .

$y \frac{d x}{d y} + y (\frac{1}{y} x) = y \cdot 2 + y \frac{1}{y^{2}}$

Step 3.2

Simplify each term.

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Step 3.2.1

Combine $\frac{1}{y}$ and $x$ .

$y \frac{d x}{d y} + y \frac{x}{y} = y \cdot 2 + y \frac{1}{y^{2}}$

Step 3.2.2

Cancel the common factor of $y$ .

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Step 3.2.2.1

Cancel the common factor.

$y \frac{d x}{d y} + y \frac{x}{y} = y \cdot 2 + y \frac{1}{y^{2}}$

Step 3.2.2.2

Rewrite the expression.

$y \frac{d x}{d y} + x = y \cdot 2 + y \frac{1}{y^{2}}$

Step 3.3

Simplify each term.

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Step 3.3.1

Move $2$ to the left of $y$ .

$y \frac{d x}{d y} + x = 2 \cdot y + y \frac{1}{y^{2}}$

Step 3.3.2

Cancel the common factor of $y$ .

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Step 3.3.2.1

Factor $y$ out of $y^{2}$ .

$y \frac{d x}{d y} + x = 2 y + y \frac{1}{y \cdot y}$

Step 3.3.2.2

Cancel the common factor.

$y \frac{d x}{d y} + x = 2 y + y \frac{1}{y \cdot y}$

Step 3.3.2.3

Rewrite the expression.

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

Step 4

Rewrite the left side as a result of differentiating a product.

$\frac{d}{d y} [y x] = 2 y + \frac{1}{y}$

Step 5

Set up an integral on each side.

$\int \frac{d}{d y} [y x] d y = \int 2 y + \frac{1}{y} d y$

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

Split the single integral into multiple integrals.

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

Step 7.2

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

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

Step 7.3

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

$y x = 2 (\frac{1}{2} y^{2} + C) + \int \frac{1}{y} d y$

Step 7.4

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

$y x = 2 (\frac{1}{2} y^{2} + C) + \ln (| y |) + C$

Step 7.5

Simplify.

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Step 7.5.1

Combine $\frac{1}{2}$ and $y^{2}$ .

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

Step 7.5.2

Simplify.

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

Step 8

Divide each term in $y x = y^{2} + \ln (| y |) + C$ by $y$ and simplify.

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Step 8.1

Divide each term in $y x = y^{2} + \ln (| y |) + C$ by $y$ .

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

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $y$ .

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Step 8.2.1.1

Cancel the common factor.

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

Step 8.2.1.2

Divide $x$ by $1$ .

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

Step 8.3

Simplify the right side.

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Step 8.3.1

Cancel the common factor of $y^{2}$ and $y$ .

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Step 8.3.1.1

Factor $y$ out of $y^{2}$ .

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

Step 8.3.1.2

Cancel the common factors.

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Step 8.3.1.2.1

Raise $y$ to the power of $1$ .

$x = \frac{y \cdot y}{y^{1}} + \frac{\ln (| y |)}{y} + \frac{C}{y}$

Step 8.3.1.2.2

Factor $y$ out of $y^{1}$ .

$x = \frac{y \cdot y}{y \cdot 1} + \frac{\ln (| y |)}{y} + \frac{C}{y}$

Step 8.3.1.2.3

Cancel the common factor.

$x = \frac{y \cdot y}{y \cdot 1} + \frac{\ln (| y |)}{y} + \frac{C}{y}$

Step 8.3.1.2.4

Rewrite the expression.

$x = \frac{y}{1} + \frac{\ln (| y |)}{y} + \frac{C}{y}$

Step 8.3.1.2.5

Divide $y$ by $1$ .

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