Calculus Examples

$2 \frac{d y}{d x} = \frac{4 d^{7} y}{d x^{2}} - 3$

Step 1

Rewrite the equation.

$2 d y = (\frac{4 d^{7} y}{d x^{2}} - 3) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int 2 d y = \int \frac{4 d^{7} y}{d x^{2}} - 3 d x$

Step 2.2

Apply the constant rule.

$2 y + C_{1} = \int \frac{4 d^{7} y}{d x^{2}} - 3 d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Apply the constant rule.

$2 y + C_{1} = (\frac{4 d^{7} y}{d x^{2}} - 3) x + C_{2}$

Step 2.3.2

Reorder terms.

$2 y + C_{1} = x (\frac{4 d^{7} y}{d x^{2}} - 3) + C_{2}$

Step 2.4

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

$2 y = x (\frac{4 d^{7} y}{d x^{2}} - 3) + K$

Step 3

Divide each term in $2 y = x (\frac{4 d^{7} y}{d x^{2}} - 3) + K$ by $2$ and simplify.

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Step 3.1

Divide each term in $2 y = x (\frac{4 d^{7} y}{d x^{2}} - 3) + K$ by $2$ .

$\frac{2 y}{2} = \frac{x (\frac{4 d^{7} y}{d x^{2}} - 3)}{2} + \frac{K}{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{x (\frac{4 d^{7} y}{d x^{2}} - 3)}{2} + \frac{K}{2}$

Step 3.2.1.2

Divide $y$ by $1$ .

$y = \frac{x (\frac{4 d^{7} y}{d x^{2}} - 3)}{2} + \frac{K}{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Simplify the numerator.

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Step 3.3.1.1.1

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{d x^{2}}{d x^{2}}$ .

$y = \frac{x (\frac{4 d^{7} y}{d x^{2}} - 3 \cdot \frac{d x^{2}}{d x^{2}})}{2} + \frac{K}{2}$

Step 3.3.1.1.2

Combine $- 3$ and $\frac{d x^{2}}{d x^{2}}$ .

$y = \frac{x (\frac{4 d^{7} y}{d x^{2}} + \frac{- 3 d x^{2}}{d x^{2}})}{2} + \frac{K}{2}$

Step 3.3.1.1.3

Combine the numerators over the common denominator.

$y = \frac{x \frac{4 d^{7} y - 3 d x^{2}}{d x^{2}}}{2} + \frac{K}{2}$

Step 3.3.1.2

Combine $x$ and $\frac{4 d^{7} y - 3 d x^{2}}{d x^{2}}$ .

$y = \frac{\frac{x (4 d^{7} y - 3 d x^{2})}{d x^{2}}}{2} + \frac{K}{2}$

Step 3.3.1.3

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{x (4 d^{7} y - 3 d x^{2})}{d x^{2}} \cdot \frac{1}{2} + \frac{K}{2}$

Step 3.3.1.4

Multiply $\frac{x (4 d^{7} y - 3 d x^{2})}{d x^{2}}$ by $\frac{1}{2}$ .

$y = \frac{x (4 d^{7} y - 3 d x^{2})}{d x^{2} \cdot 2} + \frac{K}{2}$

Step 3.3.1.5

Move $2$ to the left of $d x^{2}$ .

$y = \frac{x (4 d^{7} y - 3 d x^{2})}{2 d x^{2}} + \frac{K}{2}$

Step 3.3.2

To write $\frac{K}{2}$ as a fraction with a common denominator, multiply by $\frac{d x^{2}}{d x^{2}}$ .

$y = \frac{x (4 d^{7} y - 3 d x^{2})}{2 d x^{2}} + \frac{K}{2} \cdot \frac{d x^{2}}{d x^{2}}$

Step 3.3.3

Simplify terms.

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Step 3.3.3.1

Multiply $\frac{K}{2}$ by $\frac{d x^{2}}{d x^{2}}$ .

$y = \frac{x (4 d^{7} y - 3 d x^{2})}{2 d x^{2}} + \frac{K d x^{2}}{2 d x^{2}}$

Step 3.3.3.2

Combine the numerators over the common denominator.

$y = \frac{x (4 d^{7} y - 3 d x^{2}) + K d x^{2}}{2 d x^{2}}$

Step 3.3.4

Simplify the numerator.

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Step 3.3.4.1

Apply the distributive property.

$y = \frac{x (4 d^{7} y) + x (- 3 d x^{2}) + K d x^{2}}{2 d x^{2}}$

Step 3.3.4.2

Rewrite using the commutative property of multiplication.

$y = \frac{4 x d^{7} y + x (- 3 d x^{2}) + K d x^{2}}{2 d x^{2}}$

Step 3.3.4.3

Rewrite using the commutative property of multiplication.

$y = \frac{4 x d^{7} y - 3 x d x^{2} + K d x^{2}}{2 d x^{2}}$

Step 4

Simplify the constant of integration.

$y = \frac{4 x d^{7} y - 3 x d x^{2} + K}{2 d x^{2}}$