Calculus Examples

$\frac{d y}{d x} + 2 x y = 0$ ; dengan $y (0) = 1$

Step 1

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

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Step 1.1

Set up the integration.

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

Step 1.2

Integrate $2 x$ .

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Step 1.2.1

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

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

Step 1.2.2

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 1.2.3

Simplify the answer.

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Step 1.2.3.1

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 1.2.3.2

Simplify.

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Step 1.2.3.2.1

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

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

Step 1.2.3.2.2

Cancel the common factor of $2$ .

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Step 1.2.3.2.2.1

Cancel the common factor.

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

Step 1.2.3.2.2.2

Rewrite the expression.

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

Step 1.2.3.2.3

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

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

Step 1.3

Remove the constant of integration.

$e^{x^{2}}$

Step 2

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

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Step 2.1

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

$e^{x^{2}} \frac{d y}{d x} + e^{x^{2}} (2 x y) = e^{x^{2}} \cdot 0$

Step 2.2

Rewrite using the commutative property of multiplication.

$e^{x^{2}} \frac{d y}{d x} + 2 e^{x^{2}} (x y) = e^{x^{2}} \cdot 0$

Step 2.3

Multiply $e^{x^{2}}$ by $0$ .

$e^{x^{2}} \frac{d y}{d x} + 2 e^{x^{2}} (x y) = 0$

Step 2.4

Reorder factors in $e^{x^{2}} \frac{d y}{d x} + 2 e^{x^{2}} (x y) = 0$ .

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

Step 3

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

$\frac{d}{d x} [e^{x^{2}} y] = 0$

Step 4

Set up an integral on each side.

$\int \frac{d}{d x} [e^{x^{2}} y] d x = \int 0 d x$

Step 5

Integrate the left side.

$e^{x^{2}} y = \int 0 d x$

Step 6

Integrate the right side.

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Step 6.1

The integral of $0$ with respect to $x$ is $0$ .

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

Step 6.2

Add $0$ and $C$ .

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

Step 7

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

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Step 7.1

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

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

Step 7.2

Simplify the left side.

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Step 7.2.1

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

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Step 7.2.1.1

Cancel the common factor.

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

Step 7.2.1.2

Divide $y$ by $1$ .

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

Step 8

Use the initial condition to find the value of $C$ by substituting $0$ for $x$ and $1$ for $y$ in $y = \frac{C}{e^{x^{2}}}$ .

$1 = \frac{C}{e^{0^{2}}}$

Step 9

Solve for $C$ .

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Step 9.1

Rewrite the equation as $\frac{C}{e^{0^{2}}} = 1$ .

$\frac{C}{e^{0^{2}}} = 1$

Step 9.2

Multiply both sides of the equation by $e^{0^{2}}$ .

$e^{0^{2}} \frac{C}{e^{0^{2}}} = e^{0^{2}} \cdot 1$

Step 9.3

Simplify both sides of the equation.

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Step 9.3.1

Simplify the left side.

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Step 9.3.1.1

Cancel the common factor of $e^{0^{2}}$ .

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Step 9.3.1.1.1

Cancel the common factor.

$e^{0^{2}} \frac{C}{e^{0^{2}}} = e^{0^{2}} \cdot 1$

Step 9.3.1.1.2

Rewrite the expression.

$C = e^{0^{2}} \cdot 1$

Step 9.3.2

Simplify the right side.

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Step 9.3.2.1

Simplify $e^{0^{2}} \cdot 1$ .

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Step 9.3.2.1.1

Raising $0$ to any positive power yields $0$ .

$C = e^{0} \cdot 1$

Step 9.3.2.1.2

Anything raised to $0$ is $1$ .

$C = 1 \cdot 1$

Step 9.3.2.1.3

Multiply $1$ by $1$ .

$C = 1$

Step 10

Substitute $1$ for $C$ in $y = \frac{C}{e^{x^{2}}}$ and simplify.

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Step 10.1

Substitute $1$ for $C$ .

$y = \frac{1}{e^{x^{2}}}$