Calculus Examples

$\frac{d y}{d t} = 10 t + 9 t^{2}$ , $y (1) = 2$

Step 1

Rewrite the equation.

$d y = (10 t + 9 t^{2}) d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int 10 t + 9 t^{2} d t$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 10 t + 9 t^{2} d t$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$y + C_{1} = \int 10 t d t + \int 9 t^{2} d t$

Step 2.3.2

Since $10$ is constant with respect to $t$ , move $10$ out of the integral.

$y + C_{1} = 10 \int t d t + \int 9 t^{2} d t$

Step 2.3.3

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

$y + C_{1} = 10 (\frac{1}{2} t^{2} + C_{2}) + \int 9 t^{2} d t$

Step 2.3.4

Since $9$ is constant with respect to $t$ , move $9$ out of the integral.

$y + C_{1} = 10 (\frac{1}{2} t^{2} + C_{2}) + 9 \int t^{2} d t$

Step 2.3.5

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

$y + C_{1} = 10 (\frac{1}{2} t^{2} + C_{2}) + 9 (\frac{1}{3} t^{3} + C_{3})$

Step 2.3.6

Simplify.

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Step 2.3.6.1

Simplify.

$y + C_{1} = 10 (\frac{1}{2}) t^{2} + 9 (\frac{1}{3}) t^{3} + C_{4}$

Step 2.3.6.2

Simplify.

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Step 2.3.6.2.1

Combine $10$ and $\frac{1}{2}$ .

$y + C_{1} = \frac{10}{2} t^{2} + 9 (\frac{1}{3}) t^{3} + C_{4}$

Step 2.3.6.2.2

Cancel the common factor of $10$ and $2$ .

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Step 2.3.6.2.2.1

Factor $2$ out of $10$ .

$y + C_{1} = \frac{2 \cdot 5}{2} t^{2} + 9 (\frac{1}{3}) t^{3} + C_{4}$

Step 2.3.6.2.2.2

Cancel the common factors.

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Step 2.3.6.2.2.2.1

Factor $2$ out of $2$ .

$y + C_{1} = \frac{2 \cdot 5}{2 (1)} t^{2} + 9 (\frac{1}{3}) t^{3} + C_{4}$

Step 2.3.6.2.2.2.2

Cancel the common factor.

$y + C_{1} = \frac{2 \cdot 5}{2 \cdot 1} t^{2} + 9 (\frac{1}{3}) t^{3} + C_{4}$

Step 2.3.6.2.2.2.3

Rewrite the expression.

$y + C_{1} = \frac{5}{1} t^{2} + 9 (\frac{1}{3}) t^{3} + C_{4}$

Step 2.3.6.2.2.2.4

Divide $5$ by $1$ .

$y + C_{1} = 5 t^{2} + 9 (\frac{1}{3}) t^{3} + C_{4}$

Step 2.3.6.2.3

Combine $9$ and $\frac{1}{3}$ .

$y + C_{1} = 5 t^{2} + \frac{9}{3} t^{3} + C_{4}$

Step 2.3.6.2.4

Cancel the common factor of $9$ and $3$ .

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Step 2.3.6.2.4.1

Factor $3$ out of $9$ .

$y + C_{1} = 5 t^{2} + \frac{3 \cdot 3}{3} t^{3} + C_{4}$

Step 2.3.6.2.4.2

Cancel the common factors.

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Step 2.3.6.2.4.2.1

Factor $3$ out of $3$ .

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

Step 2.3.6.2.4.2.2

Cancel the common factor.

$y + C_{1} = 5 t^{2} + \frac{3 \cdot 3}{3 \cdot 1} t^{3} + C_{4}$

Step 2.3.6.2.4.2.3

Rewrite the expression.

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

Step 2.3.6.2.4.2.4

Divide $3$ by $1$ .

$y + C_{1} = 5 t^{2} + 3 t^{3} + C_{4}$

Step 2.3.7

Reorder terms.

$y + C_{1} = 3 t^{3} + 5 t^{2} + C_{4}$

Step 2.4

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

$y = 3 t^{3} + 5 t^{2} + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $1$ for $t$ and $2$ for $y$ in $y = 3 t^{3} + 5 t^{2} + K$ .

$2 = 3 \cdot 1^{3} + 5 \cdot 1^{2} + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $3 \cdot 1^{3} + 5 \cdot 1^{2} + K = 2$ .

$3 \cdot 1^{3} + 5 \cdot 1^{2} + K = 2$

Step 4.2

Simplify $3 \cdot 1^{3} + 5 \cdot 1^{2} + K$ .

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

One to any power is one.

$3 \cdot 1 + 5 \cdot 1^{2} + K = 2$

Step 4.2.1.2

Multiply $3$ by $1$ .

$3 + 5 \cdot 1^{2} + K = 2$

Step 4.2.1.3

One to any power is one.

$3 + 5 \cdot 1 + K = 2$

Step 4.2.1.4

Multiply $5$ by $1$ .

$3 + 5 + K = 2$

Step 4.2.2

Add $3$ and $5$ .

$8 + K = 2$

Step 4.3

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

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Step 4.3.1

Subtract $8$ from both sides of the equation.

$K = 2 - 8$

Step 4.3.2

Subtract $8$ from $2$ .

$K = - 6$

Step 5

Substitute $- 6$ for $K$ in $y = 3 t^{3} + 5 t^{2} + K$ and simplify.

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Step 5.1

Substitute $- 6$ for $K$ .

$y = 3 t^{3} + 5 t^{2} - 6$