Calculus Examples

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

Step 1

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

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Step 1.1

Apply the distributive property.

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

Step 1.2

Subtract $2 x (- y)$ from both sides of the equation.

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

Step 1.3

Reorder terms.

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

Step 1.4

Reorder terms.

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

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = - 2 \cdot - 1 x$ .

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Step 2.1

Set up the integration.

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

Step 2.2

Integrate $- 2 \cdot - 1 x$ .

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Step 2.2.1

Multiply $- 2$ by $- 1$ .

$e^{\int 2 x d x}$

Step 2.2.2

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

$e^{2 \int x d x}$

Step 2.2.3

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

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

Step 2.2.4

Simplify the answer.

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Step 2.2.4.1

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

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

Step 2.2.4.2

Simplify.

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Step 2.2.4.2.1

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

$e^{\frac{2}{2} x^{2} + C}$

Step 2.2.4.2.2

Cancel the common factor of $2$ .

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Step 2.2.4.2.2.1

Cancel the common factor.

$e^{\frac{2}{2} x^{2} + C}$

Step 2.2.4.2.2.2

Rewrite the expression.

$e^{1 x^{2} + C}$

Step 2.2.4.2.3

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

$e^{x^{2} + C}$

Step 2.3

Remove the constant of integration.

$e^{x^{2}}$

Step 3

Multiply each term by the integrating factor $e^{x^{2}}$ .

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Step 3.1

Multiply each term by $e^{x^{2}}$ .

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

Step 3.2

Simplify each term.

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Step 3.2.1

Rewrite using the commutative property of multiplication.

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

Step 3.2.2

Multiply $- 2$ by $- 1$ .

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

Step 3.3

Simplify each term.

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Step 3.3.1

Rewrite using the commutative property of multiplication.

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

Step 3.3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3.3

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 3.3.3.1

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

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Step 3.3.3.1.1

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

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

Step 3.3.3.1.2

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

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

Step 3.3.3.2

Add $2$ and $1$ .

$e^{x^{2}} \frac{d y}{d x} + 2 e^{x^{2}} (x y) = 2 e^{x^{2}} x + 2 e^{x^{2}} x^{3}$

Step 3.4

Reorder factors in $e^{x^{2}} \frac{d y}{d x} + 2 e^{x^{2}} (x y) = 2 e^{x^{2}} x + 2 e^{x^{2}} x^{3}$ .

$e^{x^{2}} \frac{d y}{d x} + 2 x y e^{x^{2}} = 2 x e^{x^{2}} + 2 x^{3} e^{x^{2}}$

Step 4

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

$\frac{d}{d x} [e^{x^{2}} y] = 2 x e^{x^{2}} + 2 x^{3} e^{x^{2}}$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [e^{x^{2}} y] d x = \int 2 x e^{x^{2}} + 2 x^{3} e^{x^{2}} d x$

Step 6

Integrate the left side.

$e^{x^{2}} y = \int 2 x e^{x^{2}} + 2 x^{3} e^{x^{2}} d x$

Step 7

Integrate the right side.

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Step 7.1

Split the single integral into multiple integrals.

$e^{x^{2}} y = \int 2 x e^{x^{2}} d x + \int 2 x^{3} e^{x^{2}} d x$

Step 7.2

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

$e^{x^{2}} y = 2 \int x e^{x^{2}} d x + \int 2 x^{3} e^{x^{2}} d x$

Step 7.3

Let $u_{2} = e^{x^{2}}$ . Then $d u_{2} = 2 x e^{x^{2}} d x$ , so $\frac{1}{2} d u_{2} = x e^{x^{2}} d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 7.3.1

Let $u_{2} = e^{x^{2}}$ . Find $\frac{d u_{2}}{d x}$ .

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Step 7.3.1.1

Differentiate $e^{x^{2}}$ .

$\frac{d}{d x} [e^{x^{2}}]$

Step 7.3.1.2

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

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Step 7.3.1.2.1

To apply the Chain Rule, set $u_{1}$ as $x^{2}$ .

$\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [x^{2}]$

Step 7.3.1.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$e^{u_{1}} \frac{d}{d x} [x^{2}]$

Step 7.3.1.2.3

Replace all occurrences of $u_{1}$ with $x^{2}$ .

$e^{x^{2}} \frac{d}{d x} [x^{2}]$

Step 7.3.1.3

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

$e^{x^{2}} (2 x)$

Step 7.3.1.4

Simplify.

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Step 7.3.1.4.1

Reorder the factors of $e^{x^{2}} (2 x)$ .

$2 e^{x^{2}} x$

Step 7.3.1.4.2

Reorder factors in $2 e^{x^{2}} x$ .

$2 x e^{x^{2}}$

Step 7.3.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$e^{x^{2}} y = 2 \int \frac{1}{2} d u_{2} + \int 2 x^{3} e^{x^{2}} d x$

Step 7.4

Apply the constant rule.

$e^{x^{2}} y = 2 (\frac{1}{2} u_{2} + C) + \int 2 x^{3} e^{x^{2}} d x$

Step 7.5

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

$e^{x^{2}} y = 2 (\frac{1}{2} u_{2} + C) + 2 \int x^{3} e^{x^{2}} d x$

Step 7.6

Let $u_{3} = x^{2}$ . Then $d u_{3} = 2 x d x$ , so $\frac{1}{2} d u_{3} = x d x$ . Rewrite using $u_{3}$ and $d$ $u_{3}$ .

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Step 7.6.1

Let $u_{3} = x^{2}$ . Find $\frac{d u_{3}}{d x}$ .

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Step 7.6.1.1

Differentiate $x^{2}$ .

$\frac{d}{d x} [x^{2}]$

Step 7.6.1.2

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

$2 x$

Step 7.6.2

Rewrite the problem using $u_{3}$ and $d u_{3}$ .

$e^{x^{2}} y = 2 (\frac{1}{2} u_{2} + C) + 2 \int u_{3} e^{u_{3}} \frac{1}{2} d u_{3}$

Step 7.7

Simplify.

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Step 7.7.1

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

$e^{x^{2}} y = 2 (\frac{u_{2}}{2} + C) + 2 \int u_{3} e^{u_{3}} \frac{1}{2} d u_{3}$

Step 7.7.2

Combine $\frac{1}{2}$ and $u_{3}$ .

$e^{x^{2}} y = 2 (\frac{u_{2}}{2} + C) + 2 \int \frac{u_{3}}{2} e^{u_{3}} d u_{3}$

Step 7.7.3

Combine $\frac{u_{3}}{2}$ and $e^{u_{3}}$ .

$e^{x^{2}} y = 2 (\frac{u_{2}}{2} + C) + 2 \int \frac{u_{3} e^{u_{3}}}{2} d u_{3}$

Step 7.8

Since $\frac{1}{2}$ is constant with respect to $u_{3}$ , move $\frac{1}{2}$ out of the integral.

$e^{x^{2}} y = 2 (\frac{u_{2}}{2} + C) + 2 (\frac{1}{2} \int u_{3} e^{u_{3}} d u_{3})$

Step 7.9

Simplify.

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Step 7.9.1

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

$e^{x^{2}} y = 2 (\frac{u_{2}}{2} + C) + \frac{2}{2} \int u_{3} e^{u_{3}} d u_{3}$

Step 7.9.2

Cancel the common factor of $2$ .

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Step 7.9.2.1

Cancel the common factor.

$e^{x^{2}} y = 2 (\frac{u_{2}}{2} + C) + \frac{2}{2} \int u_{3} e^{u_{3}} d u_{3}$

Step 7.9.2.2

Rewrite the expression.

$e^{x^{2}} y = 2 (\frac{u_{2}}{2} + C) + 1 \int u_{3} e^{u_{3}} d u_{3}$

Step 7.9.3

Multiply $\int u_{3} e^{u_{3}} d u_{3}$ by $1$ .

$e^{x^{2}} y = 2 (\frac{u_{2}}{2} + C) + \int u_{3} e^{u_{3}} d u_{3}$

Step 7.10

Integrate by parts using the formula $\int w d v = w v - \int v d w$ , where $w = u_{3}$ and $dv = e^{u_{3}}$ .

$e^{x^{2}} y = 2 (\frac{u_{2}}{2} + C) + u_{3} e^{u_{3}} - \int e^{u_{3}} d u_{3}$

Step 7.11

The integral of $e^{u_{3}}$ with respect to $u_{3}$ is $e^{u_{3}}$ .

$e^{x^{2}} y = 2 (\frac{u_{2}}{2} + C) + u_{3} e^{u_{3}} - (e^{u_{3}} + C)$

Step 7.12

Simplify.

$e^{x^{2}} y = u_{2} + u_{3} e^{u_{3}} - e^{u_{3}} + C$

Step 7.13

Substitute back in for each integration substitution variable.

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Step 7.13.1

Replace all occurrences of $u_{2}$ with $e^{x^{2}}$ .

$e^{x^{2}} y = e^{x^{2}} + u_{3} e^{u_{3}} - e^{u_{3}} + C$

Step 7.13.2

Replace all occurrences of $u_{3}$ with $x^{2}$ .

$e^{x^{2}} y = e^{x^{2}} + x^{2} e^{x^{2}} - e^{x^{2}} + C$

Step 7.14

Simplify.

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Step 7.14.1

Subtract $e^{x^{2}}$ from $e^{x^{2}}$ .

$e^{x^{2}} y = x^{2} e^{x^{2}} + 0 + C$

Step 7.14.2

Add $x^{2} e^{x^{2}}$ and $0$ .

$e^{x^{2}} y = x^{2} e^{x^{2}} + C$

Step 8

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

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Step 8.1

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

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

Step 8.2

Simplify the left side.

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Step 8.2.1

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

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Step 8.2.1.1

Cancel the common factor.

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

Step 8.2.1.2

Divide $y$ by $1$ .

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

Step 8.3

Simplify the right side.

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Step 8.3.1

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

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Step 8.3.1.1

Cancel the common factor.

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

Step 8.3.1.2

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

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