Calculus Examples

$x y y' = x^{2} + 1$

Step 1

Rewrite the differential equation.

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

Step 2

Separate the variables.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Cancel the common factor of $x$ .

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Step 2.1.2.1.1

Cancel the common factor.

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

Step 2.1.2.1.2

Rewrite the expression.

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

Step 2.1.2.2

Cancel the common factor of $y$ .

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Step 2.1.2.2.1

Cancel the common factor.

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

Step 2.1.2.2.2

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

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

Step 2.1.3

Simplify the right side.

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Step 2.1.3.1

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

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Step 2.1.3.1.1

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

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

Step 2.1.3.1.2

Cancel the common factors.

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Step 2.1.3.1.2.1

Factor $x$ out of $x y$ .

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

Step 2.1.3.1.2.2

Cancel the common factor.

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

Step 2.1.3.1.2.3

Rewrite the expression.

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

Step 2.2

Factor.

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Step 2.2.1

To write $\frac{x}{y}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 2.2.2

Write each expression with a common denominator of $y x$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.2.1

Multiply $\frac{x}{y}$ by $\frac{x}{x}$ .

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

Step 2.2.2.2

Reorder the factors of $y x$ .

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

Step 2.2.3

Combine the numerators over the common denominator.

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

Step 2.2.4

Multiply $x$ by $x$ .

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

Step 2.3

Regroup factors.

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

Step 2.4

Multiply both sides by $y$ .

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

Step 2.5

Simplify.

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Step 2.5.1

Multiply $\frac{x^{2} + 1}{x}$ by $\frac{1}{y}$ .

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

Step 2.5.2

Cancel the common factor of $y$ .

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Step 2.5.2.1

Factor $y$ out of $x y$ .

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

Step 2.5.2.2

Cancel the common factor.

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

Step 2.5.2.3

Rewrite the expression.

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

Step 2.6

Rewrite the equation.

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

Step 3

Integrate both sides.

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Step 3.1

Set up an integral on each side.

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

Step 3.2

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

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

Step 3.3

Integrate the right side.

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Step 3.3.1

Split the fraction into multiple fractions.

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

Step 3.3.2

Split the single integral into multiple integrals.

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

Step 3.3.3

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

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Step 3.3.3.1

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

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

Step 3.3.3.2

Cancel the common factors.

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Step 3.3.3.2.1

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

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

Step 3.3.3.2.2

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

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

Step 3.3.3.2.3

Cancel the common factor.

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

Step 3.3.3.2.4

Rewrite the expression.

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

Step 3.3.3.2.5

Divide $x$ by $1$ .

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

Step 3.3.4

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

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

Step 3.3.5

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

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

Step 3.3.6

Simplify.

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{2} x^{2} + \ln (| x |) + C_{4}$

Step 3.4

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

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

Step 4

Solve for $y$ .

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Step 4.1

Multiply both sides of the equation by $2$ .

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

Step 4.2

Simplify both sides of the equation.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Simplify $2 (\frac{1}{2} y^{2})$ .

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Step 4.2.1.1.1

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

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

Step 4.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.1.1.2.1

Cancel the common factor.

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

Step 4.2.1.1.2.2

Rewrite the expression.

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

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Simplify $2 (\frac{1}{2} x^{2} + \ln (| x |) + C)$ .

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Step 4.2.2.1.1

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

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

Step 4.2.2.1.2

Apply the distributive property.

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

Step 4.2.2.1.3

Cancel the common factor of $2$ .

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Step 4.2.2.1.3.1

Cancel the common factor.

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

Step 4.2.2.1.3.2

Rewrite the expression.

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

Step 4.3

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

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

Step 4.4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

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

Step 4.5

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

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

Step 4.6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.6.1

First, use the positive value of the $\pm$ to find the first solution.

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

Step 4.6.2

Next, use the negative value of the $\pm$ to find the second solution.

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

Step 4.6.3

The complete solution is the result of both the positive and negative portions of the solution.

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

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

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

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

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

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

Step 5

Simplify the constant of integration.

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

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