Calculus Examples

$\frac{d y}{d t} = 0.6 y^{\frac{1}{2}}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{y^{\frac{1}{2}}}$ .

$\frac{1}{y^{\frac{1}{2}}} \frac{d y}{d t} = \frac{1}{y^{\frac{1}{2}}} (0.6 y^{\frac{1}{2}})$

Step 1.2

Simplify.

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Step 1.2.1

Rewrite using the commutative property of multiplication.

$\frac{1}{y^{\frac{1}{2}}} \frac{d y}{d t} = 0.6 \frac{1}{y^{\frac{1}{2}}} y^{\frac{1}{2}}$

Step 1.2.2

Combine $0.6$ and $\frac{1}{y^{\frac{1}{2}}}$ .

$\frac{1}{y^{\frac{1}{2}}} \frac{d y}{d t} = \frac{0.6}{y^{\frac{1}{2}}} y^{\frac{1}{2}}$

Step 1.2.3

Cancel the common factor of $y^{\frac{1}{2}}$ .

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Step 1.2.3.1

Cancel the common factor.

$\frac{1}{y^{\frac{1}{2}}} \frac{d y}{d t} = \frac{0.6}{y^{\frac{1}{2}}} y^{\frac{1}{2}}$

Step 1.2.3.2

Rewrite the expression.

$\frac{1}{y^{\frac{1}{2}}} \frac{d y}{d t} = 0.6$

Step 1.3

Rewrite the equation.

$\frac{1}{y^{\frac{1}{2}}} d y = 0.6 d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{y^{\frac{1}{2}}} d y = \int 0.6 d t$

Step 2.2

Integrate the left side.

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Step 2.2.1

Apply basic rules of exponents.

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Step 2.2.1.1

Move $y^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

$\int {(y^{\frac{1}{2}})}^{- 1} d y = \int 0.6 d t$

Step 2.2.1.2

Multiply the exponents in ${(y^{\frac{1}{2}})}^{- 1}$ .

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Step 2.2.1.2.1

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

$\int y^{\frac{1}{2} \cdot - 1} d y = \int 0.6 d t$

Step 2.2.1.2.2

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

$\int y^{\frac{- 1}{2}} d y = \int 0.6 d t$

Step 2.2.1.2.3

Move the negative in front of the fraction.

$\int y^{- \frac{1}{2}} d y = \int 0.6 d t$

Step 2.2.2

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

$2 y^{\frac{1}{2}} + C_{1} = \int 0.6 d t$

Step 2.3

Apply the constant rule.

$2 y^{\frac{1}{2}} + C_{1} = 0.6 t + C_{2}$

Step 2.4

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

$2 y^{\frac{1}{2}} = 0.6 t + K$

Step 3

Solve for $y$ .

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Step 3.1

Divide each term in $2 y^{\frac{1}{2}} = 0.6 t + K$ by $2$ and simplify.

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Step 3.1.1

Divide each term in $2 y^{\frac{1}{2}} = 0.6 t + K$ by $2$ .

$\frac{2 y^{\frac{1}{2}}}{2} = \frac{0.6 t}{2} + \frac{K}{2}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor.

$\frac{2 y^{\frac{1}{2}}}{2} = \frac{0.6 t}{2} + \frac{K}{2}$

Step 3.1.2.2

Divide $y^{\frac{1}{2}}$ by $1$ .

$y^{\frac{1}{2}} = \frac{0.6 t}{2} + \frac{K}{2}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Simplify each term.

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Step 3.1.3.1.1

Factor $0.6$ out of $0.6 t$ .

$y^{\frac{1}{2}} = \frac{0.6 (t)}{2} + \frac{K}{2}$

Step 3.1.3.1.2

Factor $2$ out of $2$ .

$y^{\frac{1}{2}} = \frac{0.6 (t)}{2 (1)} + \frac{K}{2}$

Step 3.1.3.1.3

Separate fractions.

$y^{\frac{1}{2}} = \frac{0.6}{2} \cdot \frac{t}{1} + \frac{K}{2}$

Step 3.1.3.1.4

Divide $0.6$ by $2$ .

$y^{\frac{1}{2}} = 0.3 \frac{t}{1} + \frac{K}{2}$

Step 3.1.3.1.5

Divide $t$ by $1$ .

$y^{\frac{1}{2}} = 0.3 t + \frac{K}{2}$

Step 3.2

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${(y^{\frac{1}{2}})}^{2} = {(0.3 t + \frac{K}{2})}^{2}$

Step 3.3

Simplify the exponent.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Simplify ${(y^{\frac{1}{2}})}^{2}$ .

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Step 3.3.1.1.1

Multiply the exponents in ${(y^{\frac{1}{2}})}^{2}$ .

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Step 3.3.1.1.1.1

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

$y^{\frac{1}{2} \cdot 2} = {(0.3 t + \frac{K}{2})}^{2}$

Step 3.3.1.1.1.2

Cancel the common factor of $2$ .

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Step 3.3.1.1.1.2.1

Cancel the common factor.

$y^{\frac{1}{2} \cdot 2} = {(0.3 t + \frac{K}{2})}^{2}$

Step 3.3.1.1.1.2.2

Rewrite the expression.

$y^{1} = {(0.3 t + \frac{K}{2})}^{2}$

Step 3.3.1.1.2

Simplify.

$y = {(0.3 t + \frac{K}{2})}^{2}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify ${(0.3 t + \frac{K}{2})}^{2}$ .

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Step 3.3.2.1.1

Rewrite ${(0.3 t + \frac{K}{2})}^{2}$ as $(0.3 t + \frac{K}{2}) (0.3 t + \frac{K}{2})$ .

$y = (0.3 t + \frac{K}{2}) (0.3 t + \frac{K}{2})$

Step 3.3.2.1.2

Expand $(0.3 t + \frac{K}{2}) (0.3 t + \frac{K}{2})$ using the FOIL Method.

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Step 3.3.2.1.2.1

Apply the distributive property.

$y = 0.3 t (0.3 t + \frac{K}{2}) + \frac{K}{2} (0.3 t + \frac{K}{2})$

Step 3.3.2.1.2.2

Apply the distributive property.

$y = 0.3 t (0.3 t) + 0.3 t \frac{K}{2} + \frac{K}{2} (0.3 t + \frac{K}{2})$

Step 3.3.2.1.2.3

Apply the distributive property.

$y = 0.3 t (0.3 t) + 0.3 t \frac{K}{2} + \frac{K}{2} (0.3 t) + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3

Simplify and combine like terms.

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Step 3.3.2.1.3.1

Simplify each term.

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Step 3.3.2.1.3.1.1

Rewrite using the commutative property of multiplication.

$y = 0.3 \cdot 0.3 t \cdot t + 0.3 t \frac{K}{2} + \frac{K}{2} (0.3 t) + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.2

Multiply $t$ by $t$ by adding the exponents.

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Step 3.3.2.1.3.1.2.1

Move $t$ .

$y = 0.3 \cdot 0.3 (t \cdot t) + 0.3 t \frac{K}{2} + \frac{K}{2} (0.3 t) + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.2.2

Multiply $t$ by $t$ .

$y = 0.3 \cdot 0.3 t^{2} + 0.3 t \frac{K}{2} + \frac{K}{2} (0.3 t) + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.3

Multiply $0.3$ by $0.3$ .

$y = 0.09 t^{2} + 0.3 t \frac{K}{2} + \frac{K}{2} (0.3 t) + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.4

Multiply $0.3 t \frac{K}{2}$ .

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Step 3.3.2.1.3.1.4.1

Combine $0.3$ and $\frac{K}{2}$ .

$y = 0.09 t^{2} + t \frac{0.3 K}{2} + \frac{K}{2} (0.3 t) + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.4.2

Combine $t$ and $\frac{0.3 K}{2}$ .

$y = 0.09 t^{2} + \frac{t (0.3 K)}{2} + \frac{K}{2} (0.3 t) + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.5

Move $0.3$ to the left of $t$ .

$y = 0.09 t^{2} + \frac{0.3 \cdot t K}{2} + \frac{K}{2} (0.3 t) + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.6

Factor $0.3$ out of $0.3 \cdot t K$ .

$y = 0.09 t^{2} + \frac{0.3 (t K)}{2} + \frac{K}{2} (0.3 t) + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.7

Factor $2$ out of $2$ .

$y = 0.09 t^{2} + \frac{0.3 (t K)}{2 (1)} + \frac{K}{2} (0.3 t) + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.8

Separate fractions.

$y = 0.09 t^{2} + \frac{0.3}{2} \cdot \frac{t K}{1} + \frac{K}{2} (0.3 t) + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.9

Divide $0.3$ by $2$ .

$y = 0.09 t^{2} + 0.15 \frac{t K}{1} + \frac{K}{2} (0.3 t) + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.10

Divide $t K$ by $1$ .

$y = 0.09 t^{2} + 0.15 (t K) + \frac{K}{2} (0.3 t) + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.11

Rewrite using the commutative property of multiplication.

$y = 0.09 t^{2} + 0.15 t K + 0.3 \frac{K}{2} t + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.12

Combine $0.3$ and $\frac{K}{2}$ .

$y = 0.09 t^{2} + 0.15 t K + \frac{0.3 K}{2} t + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.13

Factor $0.3$ out of $0.3 K$ .

$y = 0.09 t^{2} + 0.15 t K + \frac{0.3 K}{2} t + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.14

Factor $2$ out of $2$ .

$y = 0.09 t^{2} + 0.15 t K + \frac{0.3 K}{2 (1)} t + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.15

Separate fractions.

$y = 0.09 t^{2} + 0.15 t K + \frac{0.3}{2} \cdot \frac{K}{1} t + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.16

Divide $0.3$ by $2$ .

$y = 0.09 t^{2} + 0.15 t K + 0.15 \frac{K}{1} t + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.17

Divide $K$ by $1$ .

$y = 0.09 t^{2} + 0.15 t K + 0.15 K t + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.18

Multiply $\frac{K}{2} \cdot \frac{K}{2}$ .

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Step 3.3.2.1.3.1.18.1

Multiply $\frac{K}{2}$ by $\frac{K}{2}$ .

$y = 0.09 t^{2} + 0.15 t K + 0.15 K t + \frac{K \cdot K}{2 \cdot 2}$

Step 3.3.2.1.3.1.18.2

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

$y = 0.09 t^{2} + 0.15 t K + 0.15 K t + \frac{K^{1} K}{2 \cdot 2}$

Step 3.3.2.1.3.1.18.3

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

$y = 0.09 t^{2} + 0.15 t K + 0.15 K t + \frac{K^{1} K^{1}}{2 \cdot 2}$

Step 3.3.2.1.3.1.18.4

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

$y = 0.09 t^{2} + 0.15 t K + 0.15 K t + \frac{K^{1 + 1}}{2 \cdot 2}$

Step 3.3.2.1.3.1.18.5

Add $1$ and $1$ .

$y = 0.09 t^{2} + 0.15 t K + 0.15 K t + \frac{K^{2}}{2 \cdot 2}$

Step 3.3.2.1.3.1.18.6

Multiply $2$ by $2$ .

$y = 0.09 t^{2} + 0.15 t K + 0.15 K t + \frac{K^{2}}{4}$

Step 3.3.2.1.3.2

Add $0.15 t K$ and $0.15 K t$ .

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Step 3.3.2.1.3.2.1

Move $t$ .

$y = 0.09 t^{2} + 0.15 K t + 0.15 K t + \frac{K^{2}}{4}$

Step 3.3.2.1.3.2.2

Add $0.15 K t$ and $0.15 K t$ .

$y = 0.09 t^{2} + 0.3 K t + \frac{K^{2}}{4}$

Step 3.4

Simplify $0.09 t^{2} + 0.3 K t + \frac{K^{2}}{4}$ .

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Step 3.4.1

Move $0.09 t^{2}$ .

$y = 0.3 K t + \frac{K^{2}}{4} + 0.09 t^{2}$

Step 3.4.2

Reorder $0.3 K t$ and $\frac{K^{2}}{4}$ .

$y = \frac{K^{2}}{4} + 0.3 K t + 0.09 t^{2}$

Step 4

Simplify the constant of integration.

$y = K + K t + 0.09 t^{2}$