Calculus Examples

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

Step 1

Let $v = y^{2}$ . Substitute $v$ for all occurrences of $y^{2}$ .

$2 x y \frac{d y}{d x} + v - 2 x = 0$

Step 2

Find $\frac{d v}{d x}$ by differentiating $y^{2}$ .

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Step 2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = y$ .

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Step 2.1.1

To apply the Chain Rule, set $u$ as $y$ .

$\frac{d v}{d x} = \frac{d}{d u} [u^{2}] \frac{d}{d x} [y]$

Step 2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

$\frac{d v}{d x} = 2 u \frac{d}{d x} [y]$

Step 2.1.3

Replace all occurrences of $u$ with $y$ .

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

Step 2.2

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$\frac{d v}{d x} = 2 y y'$

Step 3

Substitute the derivative back in to the differential equation.

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Step 3.1

Factor $2 y$ out of $2 x y$ .

$2 y (x) \frac{\frac{d v}{d x}}{2 y} + v - 2 x = 0$

Step 3.2

Cancel the common factor.

$2 y x \frac{\frac{d v}{d x}}{2 y} + v - 2 x = 0$

Step 3.3

Rewrite the expression.

$x \frac{d v}{d x} + v - 2 x = 0$

Step 4

Add $2 x$ to both sides of the equation.

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

Step 5

Check if the left side of the equation is the result of the derivative of the term $x v$ .

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Step 5.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = v$ .

$x \frac{d}{d x} [v] + v \frac{d}{d x} [x]$

Step 5.2

Rewrite $\frac{d}{d x} [v]$ as $v'$ .

$x v' + v \frac{d}{d x} [x]$

Step 5.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$x v' + v \cdot 1$

Step 5.4

Substitute $\frac{d v}{d x}$ for $v'$ .

$x \frac{d v}{d x} + v \cdot 1$

Step 5.5

Reorder $v$ and $1$ .

$x \frac{d v}{d x} + 1 \cdot v$

Step 5.6

Multiply $v$ by $1$ .

$x \frac{d v}{d x} + v$

Step 6

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

$\frac{d}{d x} [x v] = 2 x$

Step 7

Set up an integral on each side.

$\int \frac{d}{d x} [x v] d x = \int 2 x d x$

Step 8

Integrate the left side.

$x v = \int 2 x d x$

Step 9

Integrate the right side.

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Step 9.1

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

$x v = 2 \int x d x$

Step 9.2

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

$x v = 2 (\frac{1}{2} x^{2} + C)$

Step 9.3

Simplify the answer.

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Step 9.3.1

Rewrite $2 (\frac{1}{2} x^{2} + C)$ as $2 (\frac{1}{2}) x^{2} + C$ .

$x v = 2 (\frac{1}{2}) x^{2} + C$

Step 9.3.2

Simplify.

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Step 9.3.2.1

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

$x v = \frac{2}{2} x^{2} + C$

Step 9.3.2.2

Cancel the common factor of $2$ .

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Step 9.3.2.2.1

Cancel the common factor.

$x v = \frac{2}{2} x^{2} + C$

Step 9.3.2.2.2

Rewrite the expression.

$x v = 1 x^{2} + C$

Step 9.3.2.3

Multiply $x^{2}$ by $1$ .

$x v = x^{2} + C$

Step 10

Divide each term in $x v = x^{2} + C$ by $x$ and simplify.

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Step 10.1

Divide each term in $x v = x^{2} + C$ by $x$ .

$\frac{x v}{x} = \frac{x^{2}}{x} + \frac{C}{x}$

Step 10.2

Simplify the left side.

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Step 10.2.1

Cancel the common factor of $x$ .

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Step 10.2.1.1

Cancel the common factor.

$\frac{x v}{x} = \frac{x^{2}}{x} + \frac{C}{x}$

Step 10.2.1.2

Divide $v$ by $1$ .

$v = \frac{x^{2}}{x} + \frac{C}{x}$

Step 10.3

Simplify the right side.

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Step 10.3.1

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

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Step 10.3.1.1

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

$v = \frac{x \cdot x}{x} + \frac{C}{x}$

Step 10.3.1.2

Cancel the common factors.

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Step 10.3.1.2.1

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

$v = \frac{x \cdot x}{x^{1}} + \frac{C}{x}$

Step 10.3.1.2.2

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

$v = \frac{x \cdot x}{x \cdot 1} + \frac{C}{x}$

Step 10.3.1.2.3

Cancel the common factor.

$v = \frac{x \cdot x}{x \cdot 1} + \frac{C}{x}$

Step 10.3.1.2.4

Rewrite the expression.

$v = \frac{x}{1} + \frac{C}{x}$

Step 10.3.1.2.5

Divide $x$ by $1$ .

$v = x + \frac{C}{x}$

Step 11

Replace all occurrences of $v$ with $y^{2}$ .

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

Step 12

Solve for $y$ .

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Step 12.1

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

$y = \pm \sqrt{x + \frac{C}{x}}$

Step 12.2

Simplify $\pm \sqrt{x + \frac{C}{x}}$ .

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Step 12.2.1

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

$y = \pm \sqrt{\frac{x \cdot x}{x} + \frac{C}{x}}$

Step 12.2.2

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{x \cdot x + C}{x}}$

Step 12.2.3

Multiply $x$ by $x$ .

$y = \pm \sqrt{\frac{x^{2} + C}{x}}$

Step 12.2.4

Rewrite $\sqrt{\frac{x^{2} + C}{x}}$ as $\frac{\sqrt{x^{2} + C}}{\sqrt{x}}$ .

$y = \pm \frac{\sqrt{x^{2} + C}}{\sqrt{x}}$

Step 12.2.5

Multiply $\frac{\sqrt{x^{2} + C}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$y = \pm \frac{\sqrt{x^{2} + C}}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}}$

Step 12.2.6

Combine and simplify the denominator.

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Step 12.2.6.1

Multiply $\frac{\sqrt{x^{2} + C}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$y = \pm \frac{\sqrt{x^{2} + C} \sqrt{x}}{\sqrt{x} \sqrt{x}}$

Step 12.2.6.2

Raise $\sqrt{x}$ to the power of $1$ .

$y = \pm \frac{\sqrt{x^{2} + C} \sqrt{x}}{{\sqrt{x}}^{1} \sqrt{x}}$

Step 12.2.6.3

Raise $\sqrt{x}$ to the power of $1$ .

$y = \pm \frac{\sqrt{x^{2} + C} \sqrt{x}}{{\sqrt{x}}^{1} {\sqrt{x}}^{1}}$

Step 12.2.6.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \pm \frac{\sqrt{x^{2} + C} \sqrt{x}}{{\sqrt{x}}^{1 + 1}}$

Step 12.2.6.5

Add $1$ and $1$ .

$y = \pm \frac{\sqrt{x^{2} + C} \sqrt{x}}{{\sqrt{x}}^{2}}$

Step 12.2.6.6

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 12.2.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$y = \pm \frac{\sqrt{x^{2} + C} \sqrt{x}}{{(x^{\frac{1}{2}})}^{2}}$

Step 12.2.6.6.2

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

$y = \pm \frac{\sqrt{x^{2} + C} \sqrt{x}}{x^{\frac{1}{2} \cdot 2}}$

Step 12.2.6.6.3

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

$y = \pm \frac{\sqrt{x^{2} + C} \sqrt{x}}{x^{\frac{2}{2}}}$

Step 12.2.6.6.4

Cancel the common factor of $2$ .

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Step 12.2.6.6.4.1

Cancel the common factor.

$y = \pm \frac{\sqrt{x^{2} + C} \sqrt{x}}{x^{\frac{2}{2}}}$

Step 12.2.6.6.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt{x^{2} + C} \sqrt{x}}{x^{1}}$

Step 12.2.6.6.5

Simplify.

$y = \pm \frac{\sqrt{x^{2} + C} \sqrt{x}}{x}$

Step 12.2.7

Combine using the product rule for radicals.

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

Step 12.2.8

Reorder factors in $\pm \frac{\sqrt{(x^{2} + C) x}}{x}$ .

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

Step 12.3

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

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Step 12.3.1

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

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

Step 12.3.2

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

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

Step 12.3.3

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

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

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

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

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

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

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