Calculus Examples

$\frac{d y}{d x} = \frac{5 x + 3}{y - 5}$ , $y (0) = 8$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $y - 5$ .

$(y - 5) \frac{d y}{d x} = (y - 5) \frac{5 x + 3}{y - 5}$

Step 1.2

Cancel the common factor of $y - 5$ .

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Step 1.2.1

Cancel the common factor.

$(y - 5) \frac{d y}{d x} = (y - 5) \frac{5 x + 3}{y - 5}$

Step 1.2.2

Rewrite the expression.

$(y - 5) \frac{d y}{d x} = 5 x + 3$

Step 1.3

Rewrite the equation.

$(y - 5) d y = (5 x + 3) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y - 5 d y = \int 5 x + 3 d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the single integral into multiple integrals.

$\int y d y + \int - 5 d y = \int 5 x + 3 d x$

Step 2.2.2

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

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

Step 2.2.3

Apply the constant rule.

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

Step 2.2.4

Simplify.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

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

Step 2.3.3

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

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

Step 2.3.4

Apply the constant rule.

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

Step 2.3.5

Simplify.

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Step 2.3.5.1

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

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

Step 2.3.5.2

Simplify.

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

Step 2.3.5.3

Reorder terms.

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

Combine $\frac{5}{2}$ and $x^{2}$ .

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

Step 3.3

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

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Step 3.3.1

Subtract $\frac{5 x^{2}}{2}$ from both sides of the equation.

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

Step 3.3.2

Subtract $3 x$ from both sides of the equation.

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

Step 3.3.3

Subtract $K$ from both sides of the equation.

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

Step 3.4

Multiply through by the least common denominator $2$ , then simplify.

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Step 3.4.1

Apply the distributive property.

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

Step 3.4.2

Simplify.

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Step 3.4.2.1

Cancel the common factor of $2$ .

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Step 3.4.2.1.1

Cancel the common factor.

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

Step 3.4.2.1.2

Rewrite the expression.

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

Step 3.4.2.2

Multiply $- 5$ by $2$ .

$y^{2} - 10 y + 2 (- \frac{5 x^{2}}{2}) + 2 (- 3 x) + 2 (- K) = 0$

Step 3.4.2.3

Cancel the common factor of $2$ .

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Step 3.4.2.3.1

Move the leading negative in $- \frac{5 x^{2}}{2}$ into the numerator.

$y^{2} - 10 y + 2 (\frac{- 5 x^{2}}{2}) + 2 (- 3 x) + 2 (- K) = 0$

Step 3.4.2.3.2

Cancel the common factor.

$y^{2} - 10 y + 2 (\frac{- 5 x^{2}}{2}) + 2 (- 3 x) + 2 (- K) = 0$

Step 3.4.2.3.3

Rewrite the expression.

$y^{2} - 10 y - 5 x^{2} + 2 (- 3 x) + 2 (- K) = 0$

Step 3.4.2.4

Multiply $- 3$ by $2$ .

$y^{2} - 10 y - 5 x^{2} - 6 x + 2 (- K) = 0$

Step 3.4.2.5

Multiply $- 1$ by $2$ .

$y^{2} - 10 y - 5 x^{2} - 6 x - 2 K = 0$

Step 3.4.3

Move $- 10 y$ .

$y^{2} - 5 x^{2} - 6 x - 2 K - 10 y = 0$

Step 3.4.4

Move $- 6 x$ .

$y^{2} - 5 x^{2} - 2 K - 6 x - 10 y = 0$

Step 3.4.5

Reorder $y^{2}$ and $- 5 x^{2}$ .

$- 5 x^{2} + y^{2} - 2 K - 6 x - 10 y = 0$

Step 3.5

Use the quadratic formula to find the solutions.

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

Step 3.6

Substitute the values $a = 1$ , $b = - 10$ , and $c = - 5 x^{2} - 2 K - 6 x$ into the quadratic formula and solve for $y$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (1 \cdot (- 5 x^{2} - 2 K - 6 x))}}{2 \cdot 1}$

Step 3.7

Simplify.

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Step 3.7.1

Simplify the numerator.

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Step 3.7.1.1

Raise $- 10$ to the power of $2$ .

$y = \frac{10 \pm \sqrt{100 - 4 \cdot 1 \cdot (- 5 x^{2} - 2 K - 6 x)}}{2 \cdot 1}$

Step 3.7.1.2

Multiply $- 4$ by $1$ .

$y = \frac{10 \pm \sqrt{100 - 4 \cdot (- 5 x^{2} - 2 K - 6 x)}}{2 \cdot 1}$

Step 3.7.1.3

Apply the distributive property.

$y = \frac{10 \pm \sqrt{100 - 4 (- 5 x^{2}) - 4 (- 2 K) - 4 (- 6 x)}}{2 \cdot 1}$

Step 3.7.1.4

Simplify.

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Step 3.7.1.4.1

Multiply $- 5$ by $- 4$ .

$y = \frac{10 \pm \sqrt{100 + 20 x^{2} - 4 (- 2 K) - 4 (- 6 x)}}{2 \cdot 1}$

Step 3.7.1.4.2

Multiply $- 2$ by $- 4$ .

$y = \frac{10 \pm \sqrt{100 + 20 x^{2} + 8 K - 4 (- 6 x)}}{2 \cdot 1}$

Step 3.7.1.4.3

Multiply $- 6$ by $- 4$ .

$y = \frac{10 \pm \sqrt{100 + 20 x^{2} + 8 K + 24 x}}{2 \cdot 1}$

Step 3.7.1.5

Factor $4$ out of $100 + 20 x^{2} + 8 K + 24 x$ .

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Step 3.7.1.5.1

Factor $4$ out of $100$ .

$y = \frac{10 \pm \sqrt{4 (25) + 20 x^{2} + 8 K + 24 x}}{2 \cdot 1}$

Step 3.7.1.5.2

Factor $4$ out of $20 x^{2}$ .

$y = \frac{10 \pm \sqrt{4 (25) + 4 (5 x^{2}) + 8 K + 24 x}}{2 \cdot 1}$

Step 3.7.1.5.3

Factor $4$ out of $8 K$ .

$y = \frac{10 \pm \sqrt{4 (25) + 4 (5 x^{2}) + 4 (2 K) + 24 x}}{2 \cdot 1}$

Step 3.7.1.5.4

Factor $4$ out of $24 x$ .

$y = \frac{10 \pm \sqrt{4 (25) + 4 (5 x^{2}) + 4 (2 K) + 4 (6 x)}}{2 \cdot 1}$

Step 3.7.1.5.5

Factor $4$ out of $4 (25) + 4 (5 x^{2})$ .

$y = \frac{10 \pm \sqrt{4 (25 + 5 x^{2}) + 4 (2 K) + 4 (6 x)}}{2 \cdot 1}$

Step 3.7.1.5.6

Factor $4$ out of $4 (25 + 5 x^{2}) + 4 (2 K)$ .

$y = \frac{10 \pm \sqrt{4 (25 + 5 x^{2} + 2 K) + 4 (6 x)}}{2 \cdot 1}$

Step 3.7.1.5.7

Factor $4$ out of $4 (25 + 5 x^{2} + 2 K) + 4 (6 x)$ .

$y = \frac{10 \pm \sqrt{4 (25 + 5 x^{2} + 2 K + 6 x)}}{2 \cdot 1}$

Step 3.7.1.6

Rewrite $4 (25 + 5 x^{2} + 2 K + 6 x)$ as $2^{2} (5^{2} + 5 x^{2} + 2 K + 6 x)$ .

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Step 3.7.1.6.1

Rewrite $4$ as $2^{2}$ .

$y = \frac{10 \pm \sqrt{2^{2} (25 + 5 x^{2} + 2 K + 6 x)}}{2 \cdot 1}$

Step 3.7.1.6.2

Rewrite $25$ as $5^{2}$ .

$y = \frac{10 \pm \sqrt{2^{2} (5^{2} + 5 x^{2} + 2 K + 6 x)}}{2 \cdot 1}$

Step 3.7.1.7

Pull terms out from under the radical.

$y = \frac{10 \pm 2 \sqrt{5^{2} + 5 x^{2} + 2 K + 6 x}}{2 \cdot 1}$

Step 3.7.1.8

Raise $5$ to the power of $2$ .

$y = \frac{10 \pm 2 \sqrt{25 + 5 x^{2} + 2 K + 6 x}}{2 \cdot 1}$

Step 3.7.2

Multiply $2$ by $1$ .

$y = \frac{10 \pm 2 \sqrt{25 + 5 x^{2} + 2 K + 6 x}}{2}$

Step 3.7.3

Simplify $\frac{10 \pm 2 \sqrt{25 + 5 x^{2} + 2 K + 6 x}}{2}$ .

$y = 5 \pm \sqrt{25 + 5 x^{2} + 2 K + 6 x}$

Step 3.8

The final answer is the combination of both solutions.

$y = 5 + \sqrt{25 + 5 x^{2} + 2 K + 6 x}$

$y = 5 - \sqrt{25 + 5 x^{2} + 2 K + 6 x}$

$y = 5 + \sqrt{25 + 5 x^{2} + 2 K + 6 x}$

$y = 5 - \sqrt{25 + 5 x^{2} + 2 K + 6 x}$

Step 4

Simplify the constant of integration.

$y = 5 + \sqrt{25 + 5 x^{2} + K + 6 x}$

$y = 5 - \sqrt{25 + 5 x^{2} + K + 6 x}$

Step 5

Since $y$ is positive in the initial condition $(0, 8)$ , only consider $y = 5 + \sqrt{25 + 5 x^{2} + K + 6 x}$ to find the $K$ . Substitute $0$ for $x$ and $8$ for $y$ .

$8 = 5 + \sqrt{25 + 5 \cdot 0^{2} + K + 6 \cdot 0}$

Step 6

Solve for $K$ .

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Step 6.1

Rewrite the equation as $5 + \sqrt{25 + 5 \cdot 0^{2} + K + 6 \cdot 0} = 8$ .

$5 + \sqrt{25 + 5 \cdot 0^{2} + K + 6 \cdot 0} = 8$

Step 6.2

Move all terms not containing $\sqrt{25 + 5 \cdot 0^{2} + K + 6 \cdot 0}$ to the right side of the equation.

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Step 6.2.1

Subtract $5$ from both sides of the equation.

$\sqrt{25 + 5 \cdot 0^{2} + K + 6 \cdot 0} = 8 - 5$

Step 6.2.2

Subtract $5$ from $8$ .

$\sqrt{25 + 5 \cdot 0^{2} + K + 6 \cdot 0} = 3$

Step 6.3

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

${\sqrt{25 + 5 \cdot 0^{2} + K + 6 \cdot 0}}^{2} = 3^{2}$

Step 6.4

Simplify each side of the equation.

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Step 6.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{25 + 5 \cdot 0^{2} + K + 6 \cdot 0}$ as ${(25 + 5 \cdot 0^{2} + K + 6 \cdot 0)}^{\frac{1}{2}}$ .

${({(25 + 5 \cdot 0^{2} + K + 6 \cdot 0)}^{\frac{1}{2}})}^{2} = 3^{2}$

Step 6.4.2

Simplify the left side.

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Step 6.4.2.1

Simplify ${({(25 + 5 \cdot 0^{2} + K + 6 \cdot 0)}^{\frac{1}{2}})}^{2}$ .

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Step 6.4.2.1.1

Multiply the exponents in ${({(25 + 5 \cdot 0^{2} + K + 6 \cdot 0)}^{\frac{1}{2}})}^{2}$ .

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Step 6.4.2.1.1.1

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

${(25 + 5 \cdot 0^{2} + K + 6 \cdot 0)}^{\frac{1}{2} \cdot 2} = 3^{2}$

Step 6.4.2.1.1.2

Cancel the common factor of $2$ .

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Step 6.4.2.1.1.2.1

Cancel the common factor.

${(25 + 5 \cdot 0^{2} + K + 6 \cdot 0)}^{\frac{1}{2} \cdot 2} = 3^{2}$

Step 6.4.2.1.1.2.2

Rewrite the expression.

${(25 + 5 \cdot 0^{2} + K + 6 \cdot 0)}^{1} = 3^{2}$

Step 6.4.2.1.2

Simplify each term.

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Step 6.4.2.1.2.1

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

${(25 + 5 \cdot 0 + K + 6 \cdot 0)}^{1} = 3^{2}$

Step 6.4.2.1.2.2

Multiply $5$ by $0$ .

${(25 + 0 + K + 6 \cdot 0)}^{1} = 3^{2}$

Step 6.4.2.1.2.3

Multiply $6$ by $0$ .

${(25 + 0 + K + 0)}^{1} = 3^{2}$

Step 6.4.2.1.3

Simplify by adding terms.

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Step 6.4.2.1.3.1

Combine the opposite terms in $25 + 0 + K + 0$ .

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Step 6.4.2.1.3.1.1

Add $25$ and $0$ .

${(25 + K + 0)}^{1} = 3^{2}$

Step 6.4.2.1.3.1.2

Add $25 + K$ and $0$ .

${(25 + K)}^{1} = 3^{2}$

Step 6.4.2.1.3.2

Simplify.

$25 + K = 3^{2}$

Step 6.4.3

Simplify the right side.

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Step 6.4.3.1

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

$25 + K = 9$

Step 6.5

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

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Step 6.5.1

Subtract $25$ from both sides of the equation.

$K = 9 - 25$

Step 6.5.2

Subtract $25$ from $9$ .

$K = - 16$

Step 7

Substitute $- 16$ for $K$ in $y = 5 + \sqrt{25 + 5 x^{2} + K + 6 x}$ and simplify.

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Step 7.1

Substitute $- 16$ for $K$ .

$y = 5 + \sqrt{25 + 5 x^{2} - 16 + 6 x}$

Step 7.2

Simplify each term.

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Step 7.2.1

Subtract $16$ from $25$ .

$y = 5 + \sqrt{5 x^{2} + 9 + 6 x}$

Step 7.2.2

Reorder terms.

$y = 5 + \sqrt{5 x^{2} + 6 x + 9}$