Calculus Examples

$\frac{d y}{d t} = \frac{y}{t}$

Step 1

Let $V = \frac{y}{t}$ . Substitute $V$ for $\frac{y}{t}$ .

$\frac{d y}{d t} = V$

Step 2

Solve $V = \frac{y}{t}$ for $y$ .

$y = V t$

Step 3

Use the product rule to find the derivative of $y = V t$ with respect to $t$ .

$\frac{d y}{d t} = t \frac{d V}{d t} + V$

Step 4

Substitute $t \frac{d V}{d t} + V$ for $\frac{d y}{d t}$ .

$t \frac{d V}{d t} + V = V$

Step 5

Solve the substituted differential equation.

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Step 5.1

Separate the variables.

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Step 5.1.1

Solve for $\frac{d V}{d t}$ .

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Step 5.1.1.1

Move all terms not containing $\frac{d V}{d t}$ to the right side of the equation.

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Step 5.1.1.1.1

Subtract $V$ from both sides of the equation.

$t \frac{d V}{d t} = V - V$

Step 5.1.1.1.2

Subtract $V$ from $V$ .

$t \frac{d V}{d t} = 0$

Step 5.1.1.2

Divide each term in $t \frac{d V}{d t} = 0$ by $t$ and simplify.

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Step 5.1.1.2.1

Divide each term in $t \frac{d V}{d t} = 0$ by $t$ .

$\frac{t \frac{d V}{d t}}{t} = \frac{0}{t}$

Step 5.1.1.2.2

Simplify the left side.

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Step 5.1.1.2.2.1

Cancel the common factor of $t$ .

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Step 5.1.1.2.2.1.1

Cancel the common factor.

$\frac{t \frac{d V}{d t}}{t} = \frac{0}{t}$

Step 5.1.1.2.2.1.2

Divide $\frac{d V}{d t}$ by $1$ .

$\frac{d V}{d t} = \frac{0}{t}$

Step 5.1.1.2.3

Simplify the right side.

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Step 5.1.1.2.3.1

Divide $0$ by $t$ .

$\frac{d V}{d t} = 0$

Step 5.1.2

Rewrite the equation.

$d V = 0 d t$

Step 5.2

Integrate both sides.

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Step 5.2.1

Set up an integral on each side.

$\int d V = \int 0 d t$

Step 5.2.2

Apply the constant rule.

$V + C_{1} = \int 0 d t$

Step 5.2.3

Integrate the right side.

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Step 5.2.3.1

The integral of $0$ with respect to $t$ is $0$ .

$V + C_{1} = 0 + C_{2}$

Step 5.2.3.2

Add $0$ and $C_{2}$ .

$V + C_{1} = C_{2}$

Step 5.2.4

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

$V = D$

Step 6

Substitute $\frac{y}{t}$ for $V$ .

$\frac{y}{t} = D$

Step 7

Solve $\frac{y}{t} = D$ for $y$ .

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Step 7.1

Multiply both sides by $t$ .

$\frac{y}{t} t = D t$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $t$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{y}{t} t = D t$

Step 7.2.1.2

Rewrite the expression.

$y = D t$