Calculus Examples

$x \frac{d y}{d x} + 2 y = 4 x^{2}$ , $y (1) = 0$

Step 1

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

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Step 1.1

Divide each term in $x \frac{d y}{d x} + 2 y = 4 x^{2}$ by $x$ .

$\frac{x \frac{d y}{d x}}{x} + \frac{2 y}{x} = \frac{4 x^{2}}{x}$

Step 1.2

Cancel the common factor of $x$ .

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Step 1.2.1

Cancel the common factor.

$\frac{x \frac{d y}{d x}}{x} + \frac{2 y}{x} = \frac{4 x^{2}}{x}$

Step 1.2.2

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

$\frac{d y}{d x} + \frac{2 y}{x} = \frac{4 x^{2}}{x}$

Step 1.3

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

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Step 1.3.1

Factor $x$ out of $4 x^{2}$ .

$\frac{d y}{d x} + \frac{2 y}{x} = \frac{x (4 x)}{x}$

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

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

$\frac{d y}{d x} + \frac{2 y}{x} = \frac{x (4 x)}{x^{1}}$

Step 1.3.2.2

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

$\frac{d y}{d x} + \frac{2 y}{x} = \frac{x (4 x)}{x \cdot 1}$

Step 1.3.2.3

Cancel the common factor.

$\frac{d y}{d x} + \frac{2 y}{x} = \frac{x (4 x)}{x \cdot 1}$

Step 1.3.2.4

Rewrite the expression.

$\frac{d y}{d x} + \frac{2 y}{x} = \frac{4 x}{1}$

Step 1.3.2.5

Divide $4 x$ by $1$ .

$\frac{d y}{d x} + \frac{2 y}{x} = 4 x$

Step 1.4

Factor $y$ out of $\frac{2 y}{x}$ .

$\frac{d y}{d x} + y \frac{2}{x} = 4 x$

Step 1.5

Reorder $y$ and $\frac{2}{x}$ .

$\frac{d y}{d x} + \frac{2}{x} y = 4 x$

Step 2

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

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Step 2.1

Set up the integration.

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

Step 2.2

Integrate $\frac{2}{x}$ .

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Step 2.2.1

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

$e^{2 \int \frac{1}{x} d x}$

Step 2.2.2

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

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

Step 2.2.3

Simplify.

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

Step 2.3

Remove the constant of integration.

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

Step 2.4

Use the logarithmic power rule.

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

Step 2.5

Exponentiation and log are inverse functions.

$x^{2}$

Step 3

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

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Step 3.1

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

$x^{2} \frac{d y}{d x} + x^{2} (\frac{2}{x} y) = x^{2} (4 x)$

Step 3.2

Simplify each term.

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Step 3.2.1

Combine $\frac{2}{x}$ and $y$ .

$x^{2} \frac{d y}{d x} + x^{2} \frac{2 y}{x} = x^{2} (4 x)$

Step 3.2.2

Cancel the common factor of $x$ .

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Step 3.2.2.1

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

$x^{2} \frac{d y}{d x} + x \cdot x \frac{2 y}{x} = x^{2} (4 x)$

Step 3.2.2.2

Cancel the common factor.

$x^{2} \frac{d y}{d x} + x \cdot x \frac{2 y}{x} = x^{2} (4 x)$

Step 3.2.2.3

Rewrite the expression.

$x^{2} \frac{d y}{d x} + x (2 y) = x^{2} (4 x)$

Step 3.2.3

Rewrite using the commutative property of multiplication.

$x^{2} \frac{d y}{d x} + 2 x y = x^{2} (4 x)$

Step 3.3

Rewrite using the commutative property of multiplication.

$x^{2} \frac{d y}{d x} + 2 x y = 4 x^{2} x$

Step 3.4

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

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Step 3.4.1

Move $x$ .

$x^{2} \frac{d y}{d x} + 2 x y = 4 (x \cdot x^{2})$

Step 3.4.2

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

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Step 3.4.2.1

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

$x^{2} \frac{d y}{d x} + 2 x y = 4 (x^{1} x^{2})$

Step 3.4.2.2

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

$x^{2} \frac{d y}{d x} + 2 x y = 4 x^{1 + 2}$

Step 3.4.3

Add $1$ and $2$ .

$x^{2} \frac{d y}{d x} + 2 x y = 4 x^{3}$

Step 4

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

$\frac{d}{d x} [x^{2} y] = 4 x^{3}$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [x^{2} y] d x = \int 4 x^{3} d x$

Step 6

Integrate the left side.

$x^{2} y = \int 4 x^{3} d x$

Step 7

Integrate the right side.

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Step 7.1

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$x^{2} y = 4 \int x^{3} d x$

Step 7.2

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$x^{2} y = 4 (\frac{1}{4} x^{4} + C)$

Step 7.3

Simplify the answer.

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Step 7.3.1

Rewrite $4 (\frac{1}{4} x^{4} + C)$ as $4 (\frac{1}{4}) x^{4} + C$ .

$x^{2} y = 4 (\frac{1}{4}) x^{4} + C$

Step 7.3.2

Simplify.

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Step 7.3.2.1

Combine $4$ and $\frac{1}{4}$ .

$x^{2} y = \frac{4}{4} x^{4} + C$

Step 7.3.2.2

Cancel the common factor of $4$ .

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Step 7.3.2.2.1

Cancel the common factor.

$x^{2} y = \frac{4}{4} x^{4} + C$

Step 7.3.2.2.2

Rewrite the expression.

$x^{2} y = 1 x^{4} + C$

Step 7.3.2.3

Multiply $x^{4}$ by $1$ .

$x^{2} y = x^{4} + C$

Step 8

Divide each term in $x^{2} y = x^{4} + C$ by $x^{2}$ and simplify.

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Step 8.1

Divide each term in $x^{2} y = x^{4} + C$ by $x^{2}$ .

$\frac{x^{2} y}{x^{2}} = \frac{x^{4}}{x^{2}} + \frac{C}{x^{2}}$

Step 8.2

Simplify the left side.

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Step 8.2.1

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

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Step 8.2.1.1

Cancel the common factor.

$\frac{x^{2} y}{x^{2}} = \frac{x^{4}}{x^{2}} + \frac{C}{x^{2}}$

Step 8.2.1.2

Divide $y$ by $1$ .

$y = \frac{x^{4}}{x^{2}} + \frac{C}{x^{2}}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Cancel the common factor of $x^{4}$ and $x^{2}$ .

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Step 8.3.1.1

Factor $x^{2}$ out of $x^{4}$ .

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

Step 8.3.1.2

Cancel the common factors.

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Step 8.3.1.2.1

Multiply by $1$ .

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

Step 8.3.1.2.2

Cancel the common factor.

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

Step 8.3.1.2.3

Rewrite the expression.

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

Step 8.3.1.2.4

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

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

Step 9

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

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

Step 10

Solve for $C$ .

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Step 10.1

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

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

Step 10.2

Simplify each term.

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Step 10.2.1

One to any power is one.

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

Step 10.2.2

One to any power is one.

$1 + \frac{C}{1} = 0$

Step 10.2.3

Divide $C$ by $1$ .

$1 + C = 0$

Step 10.3

Subtract $1$ from both sides of the equation.

$C = - 1$

Step 11

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

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Step 11.1

Substitute $- 1$ for $C$ .

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

Step 11.2

Move the negative in front of the fraction.

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