Calculus Examples

$\frac{d y}{d x} = \frac{x^{2} + 5}{2 y - 1}$ , $y (0) = 11$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $2 y - 1$ .

$(2 y - 1) \frac{d y}{d x} = (2 y - 1) \frac{x^{2} + 5}{2 y - 1}$

Step 1.2

Cancel the common factor of $2 y - 1$ .

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Step 1.2.1

Cancel the common factor.

$(2 y - 1) \frac{d y}{d x} = (2 y - 1) \frac{x^{2} + 5}{2 y - 1}$

Step 1.2.2

Rewrite the expression.

$(2 y - 1) \frac{d y}{d x} = x^{2} + 5$

Step 1.3

Rewrite the equation.

$(2 y - 1) d y = (x^{2} + 5) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int 2 y - 1 d y = \int x^{2} + 5 d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the single integral into multiple integrals.

$\int 2 y d y + \int - 1 d y = \int x^{2} + 5 d x$

Step 2.2.2

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

$2 \int y d y + \int - 1 d y = \int x^{2} + 5 d x$

Step 2.2.3

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

$2 (\frac{1}{2} y^{2} + C_{1}) + \int - 1 d y = \int x^{2} + 5 d x$

Step 2.2.4

Apply the constant rule.

$2 (\frac{1}{2} y^{2} + C_{1}) - y + C_{2} = \int x^{2} + 5 d x$

Step 2.2.5

Simplify.

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Step 2.2.5.1

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

$2 (\frac{y^{2}}{2} + C_{1}) - y + C_{2} = \int x^{2} + 5 d x$

Step 2.2.5.2

Simplify.

$y^{2} - y + C_{3} = \int x^{2} + 5 d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$y^{2} - y + C_{3} = \int x^{2} d x + \int 5 d x$

Step 2.3.2

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$y^{2} - y + C_{3} = \frac{1}{3} x^{3} + C_{4} + \int 5 d x$

Step 2.3.3

Apply the constant rule.

$y^{2} - y + C_{3} = \frac{1}{3} x^{3} + C_{4} + 5 x + C_{5}$

Step 2.3.4

Simplify.

$y^{2} - y + C_{3} = \frac{1}{3} x^{3} + 5 x + C_{6}$

Step 2.4

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

$y^{2} - y = \frac{1}{3} x^{3} + 5 x + K$

Step 3

Solve for $y$ .

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Step 3.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$y^{2} - y = \frac{x^{3}}{3} + 5 x + K$

Step 3.2

Move all the expressions to the left side of the equation.

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Step 3.2.1

Subtract $\frac{x^{3}}{3}$ from both sides of the equation.

$y^{2} - y - \frac{x^{3}}{3} = 5 x + K$

Step 3.2.2

Subtract $5 x$ from both sides of the equation.

$y^{2} - y - \frac{x^{3}}{3} - 5 x = K$

Step 3.2.3

Subtract $K$ from both sides of the equation.

$y^{2} - y - \frac{x^{3}}{3} - 5 x - K = 0$

Step 3.3

Multiply through by the least common denominator $3$ , then simplify.

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Step 3.3.1

Apply the distributive property.

$3 y^{2} + 3 (- y) + 3 (- \frac{x^{3}}{3}) + 3 (- 5 x) + 3 (- K) = 0$

Step 3.3.2

Simplify.

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Step 3.3.2.1

Multiply $- 1$ by $3$ .

$3 y^{2} - 3 y + 3 (- \frac{x^{3}}{3}) + 3 (- 5 x) + 3 (- K) = 0$

Step 3.3.2.2

Cancel the common factor of $3$ .

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Step 3.3.2.2.1

Move the leading negative in $- \frac{x^{3}}{3}$ into the numerator.

$3 y^{2} - 3 y + 3 (\frac{- x^{3}}{3}) + 3 (- 5 x) + 3 (- K) = 0$

Step 3.3.2.2.2

Cancel the common factor.

$3 y^{2} - 3 y + 3 (\frac{- x^{3}}{3}) + 3 (- 5 x) + 3 (- K) = 0$

Step 3.3.2.2.3

Rewrite the expression.

$3 y^{2} - 3 y - x^{3} + 3 (- 5 x) + 3 (- K) = 0$

Step 3.3.2.3

Multiply $- 5$ by $3$ .

$3 y^{2} - 3 y - x^{3} - 15 x + 3 (- K) = 0$

Step 3.3.2.4

Multiply $- 1$ by $3$ .

$3 y^{2} - 3 y - x^{3} - 15 x - 3 K = 0$

Step 3.3.3

Move $- 15 x$ .

$3 y^{2} - 3 y - x^{3} - 3 K - 15 x = 0$

Step 3.3.4

Move $- 3 y$ .

$3 y^{2} - x^{3} - 3 K - 3 y - 15 x = 0$

Step 3.3.5

Reorder $3 y^{2}$ and $- x^{3}$ .

$- x^{3} + 3 y^{2} - 3 K - 3 y - 15 x = 0$

Step 3.4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3.5

Substitute the values $a = 3$ , $b = - 3$ , and $c = - x^{3} - 3 K - 15 x$ into the quadratic formula and solve for $y$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (3 \cdot (- x^{3} - 3 K - 15 x))}}{2 \cdot 3}$

Step 3.6

Simplify.

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

Raise $- 3$ to the power of $2$ .

$y = \frac{3 \pm \sqrt{9 - 4 \cdot 3 \cdot (- x^{3} - 3 K - 15 x)}}{2 \cdot 3}$

Step 3.6.1.2

Multiply $- 4$ by $3$ .

$y = \frac{3 \pm \sqrt{9 - 12 \cdot (- x^{3} - 3 K - 15 x)}}{2 \cdot 3}$

Step 3.6.1.3

Apply the distributive property.

$y = \frac{3 \pm \sqrt{9 - 12 (- x^{3}) - 12 (- 3 K) - 12 (- 15 x)}}{2 \cdot 3}$

Step 3.6.1.4

Simplify.

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Step 3.6.1.4.1

Multiply $- 1$ by $- 12$ .

$y = \frac{3 \pm \sqrt{9 + 12 x^{3} - 12 (- 3 K) - 12 (- 15 x)}}{2 \cdot 3}$

Step 3.6.1.4.2

Multiply $- 3$ by $- 12$ .

$y = \frac{3 \pm \sqrt{9 + 12 x^{3} + 36 K - 12 (- 15 x)}}{2 \cdot 3}$

Step 3.6.1.4.3

Multiply $- 15$ by $- 12$ .

$y = \frac{3 \pm \sqrt{9 + 12 x^{3} + 36 K + 180 x}}{2 \cdot 3}$

Step 3.6.1.5

Factor $3$ out of $9 + 12 x^{3} + 36 K + 180 x$ .

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Step 3.6.1.5.1

Factor $3$ out of $9$ .

$y = \frac{3 \pm \sqrt{3 (3) + 12 x^{3} + 36 K + 180 x}}{2 \cdot 3}$

Step 3.6.1.5.2

Factor $3$ out of $12 x^{3}$ .

$y = \frac{3 \pm \sqrt{3 (3) + 3 (4 x^{3}) + 36 K + 180 x}}{2 \cdot 3}$

Step 3.6.1.5.3

Factor $3$ out of $36 K$ .

$y = \frac{3 \pm \sqrt{3 (3) + 3 (4 x^{3}) + 3 (12 K) + 180 x}}{2 \cdot 3}$

Step 3.6.1.5.4

Factor $3$ out of $180 x$ .

$y = \frac{3 \pm \sqrt{3 (3) + 3 (4 x^{3}) + 3 (12 K) + 3 (60 x)}}{2 \cdot 3}$

Step 3.6.1.5.5

Factor $3$ out of $3 (3) + 3 (4 x^{3})$ .

$y = \frac{3 \pm \sqrt{3 (3 + 4 x^{3}) + 3 (12 K) + 3 (60 x)}}{2 \cdot 3}$

Step 3.6.1.5.6

Factor $3$ out of $3 (3 + 4 x^{3}) + 3 (12 K)$ .

$y = \frac{3 \pm \sqrt{3 (3 + 4 x^{3} + 12 K) + 3 (60 x)}}{2 \cdot 3}$

Step 3.6.1.5.7

Factor $3$ out of $3 (3 + 4 x^{3} + 12 K) + 3 (60 x)$ .

$y = \frac{3 \pm \sqrt{3 (3 + 4 x^{3} + 12 K + 60 x)}}{2 \cdot 3}$

Step 3.6.2

Multiply $2$ by $3$ .

$y = \frac{3 \pm \sqrt{3 (3 + 4 x^{3} + 12 K + 60 x)}}{6}$

Step 3.7

The final answer is the combination of both solutions.

$y = \frac{3 + \sqrt{3 (3 + 4 x^{3} + 12 K + 60 x)}}{6}$

$y = \frac{3 - \sqrt{3 (3 + 4 x^{3} + 12 K + 60 x)}}{6}$

$y = \frac{3 + \sqrt{3 (3 + 4 x^{3} + 12 K + 60 x)}}{6}$

$y = \frac{3 - \sqrt{3 (3 + 4 x^{3} + 12 K + 60 x)}}{6}$

Step 4

Simplify the constant of integration.

$y = \frac{3 + \sqrt{3 (3 + 4 x^{3} + K + 60 x)}}{6}$

$y = \frac{3 - \sqrt{3 (3 + 4 x^{3} + K + 60 x)}}{6}$

Step 5

Since $y$ is positive in the initial condition $(0, 11)$ , only consider $y = \frac{3 + \sqrt{3 (3 + 4 x^{3} + K + 60 x)}}{6}$ to find the $K$ . Substitute $0$ for $x$ and $11$ for $y$ .

$11 = \frac{3 + \sqrt{3 (3 + 4 \cdot 0^{3} + K + 60 \cdot 0)}}{6}$

Step 6

Solve for $K$ .

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Step 6.1

Rewrite the equation as $\frac{3 + \sqrt{3 (3 + 4 \cdot 0^{3} + K + 60 \cdot 0)}}{6} = 11$ .

$\frac{3 + \sqrt{3 (3 + 4 \cdot 0^{3} + K + 60 \cdot 0)}}{6} = 11$

Step 6.2

Multiply both sides by $6$ .

$\frac{3 + \sqrt{3 (3 + 4 \cdot 0^{3} + K + 60 \cdot 0)}}{6} \cdot 6 = 11 \cdot 6$

Step 6.3

Simplify.

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Step 6.3.1

Simplify the left side.

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Step 6.3.1.1

Simplify $\frac{3 + \sqrt{3 (3 + 4 \cdot 0^{3} + K + 60 \cdot 0)}}{6} \cdot 6$ .

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Step 6.3.1.1.1

Simplify the numerator.

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Step 6.3.1.1.1.1

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

$\frac{3 + \sqrt{3 (3 + 4 \cdot 0 + K + 60 \cdot 0)}}{6} \cdot 6 = 11 \cdot 6$

Step 6.3.1.1.1.2

Multiply $4$ by $0$ .

$\frac{3 + \sqrt{3 (3 + 0 + K + 60 \cdot 0)}}{6} \cdot 6 = 11 \cdot 6$

Step 6.3.1.1.1.3

Multiply $60$ by $0$ .

$\frac{3 + \sqrt{3 (3 + 0 + K + 0)}}{6} \cdot 6 = 11 \cdot 6$

Step 6.3.1.1.1.4

Add $3 + 0 + K$ and $0$ .

$\frac{3 + \sqrt{3 (3 + 0 + K)}}{6} \cdot 6 = 11 \cdot 6$

Step 6.3.1.1.1.5

Add $3$ and $0$ .

$\frac{3 + \sqrt{3 (3 + K)}}{6} \cdot 6 = 11 \cdot 6$

Step 6.3.1.1.2

Reduce the expression by cancelling the common factors.

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Step 6.3.1.1.2.1

Cancel the common factor of $6$ .

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Step 6.3.1.1.2.1.1

Cancel the common factor.

$\frac{3 + \sqrt{3 (3 + K)}}{6} \cdot 6 = 11 \cdot 6$

Step 6.3.1.1.2.1.2

Rewrite the expression.

$3 + \sqrt{3 (3 + K)} = 11 \cdot 6$

Step 6.3.1.1.2.2

Reorder $3$ and $K$ .

$3 + \sqrt{3 (K + 3)} = 11 \cdot 6$

Step 6.3.2

Simplify the right side.

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Step 6.3.2.1

Multiply $11$ by $6$ .

$3 + \sqrt{3 (K + 3)} = 66$

Step 6.4

Solve for $K$ .

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Step 6.4.1

Move all terms not containing $\sqrt{3 (K + 3)}$ to the right side of the equation.

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Step 6.4.1.1

Subtract $3$ from both sides of the equation.

$\sqrt{3 (K + 3)} = 66 - 3$

Step 6.4.1.2

Subtract $3$ from $66$ .

$\sqrt{3 (K + 3)} = 63$

Step 6.4.2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{3 (K + 3)}}^{2} = 63^{2}$

Step 6.4.3

Simplify each side of the equation.

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Step 6.4.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 (K + 3)}$ as ${(3 (K + 3))}^{\frac{1}{2}}$ .

${({(3 (K + 3))}^{\frac{1}{2}})}^{2} = 63^{2}$

Step 6.4.3.2

Simplify the left side.

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Step 6.4.3.2.1

Simplify ${({(3 (K + 3))}^{\frac{1}{2}})}^{2}$ .

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Step 6.4.3.2.1.1

Multiply the exponents in ${({(3 (K + 3))}^{\frac{1}{2}})}^{2}$ .

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Step 6.4.3.2.1.1.1

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

${(3 (K + 3))}^{\frac{1}{2} \cdot 2} = 63^{2}$

Step 6.4.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 6.4.3.2.1.1.2.1

Cancel the common factor.

${(3 (K + 3))}^{\frac{1}{2} \cdot 2} = 63^{2}$

Step 6.4.3.2.1.1.2.2

Rewrite the expression.

${(3 (K + 3))}^{1} = 63^{2}$

Step 6.4.3.2.1.2

Apply the distributive property.

${(3 K + 3 \cdot 3)}^{1} = 63^{2}$

Step 6.4.3.2.1.3

Multiply.

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Step 6.4.3.2.1.3.1

Multiply $3$ by $3$ .

${(3 K + 9)}^{1} = 63^{2}$

Step 6.4.3.2.1.3.2

Simplify.

$3 K + 9 = 63^{2}$

Step 6.4.3.3

Simplify the right side.

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Step 6.4.3.3.1

Raise $63$ to the power of $2$ .

$3 K + 9 = 3969$

Step 6.4.4

Solve for $K$ .

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Step 6.4.4.1

Move all terms not containing $K$ to the right side of the equation.

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Step 6.4.4.1.1

Subtract $9$ from both sides of the equation.

$3 K = 3969 - 9$

Step 6.4.4.1.2

Subtract $9$ from $3969$ .

$3 K = 3960$

Step 6.4.4.2

Divide each term in $3 K = 3960$ by $3$ and simplify.

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Step 6.4.4.2.1

Divide each term in $3 K = 3960$ by $3$ .

$\frac{3 K}{3} = \frac{3960}{3}$

Step 6.4.4.2.2

Simplify the left side.

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Step 6.4.4.2.2.1

Cancel the common factor of $3$ .

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Step 6.4.4.2.2.1.1

Cancel the common factor.

$\frac{3 K}{3} = \frac{3960}{3}$

Step 6.4.4.2.2.1.2

Divide $K$ by $1$ .

$K = \frac{3960}{3}$

Step 6.4.4.2.3

Simplify the right side.

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Step 6.4.4.2.3.1

Divide $3960$ by $3$ .

$K = 1320$

Step 7

Substitute $1320$ for $K$ in $y = \frac{3 + \sqrt{3 (3 + 4 x^{3} + K + 60 x)}}{6}$ and simplify.

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Step 7.1

Substitute $1320$ for $K$ .

$y = \frac{3 + \sqrt{3 (3 + 4 x^{3} + 1320 + 60 x)}}{6}$

Step 7.2

Simplify the numerator.

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Step 7.2.1

Add $3$ and $1320$ .

$y = \frac{3 + \sqrt{3 (4 x^{3} + 1323 + 60 x)}}{6}$

Step 7.2.2

Reorder terms.

$y = \frac{3 + \sqrt{3 (4 x^{3} + 60 x + 1323)}}{6}$