Calculus Examples

$\frac{d y}{d t} = t^{\frac{5}{2}} y^{4}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{y^{4}}$ .

$\frac{1}{y^{4}} \frac{d y}{d t} = \frac{1}{y^{4}} (t^{\frac{5}{2}} y^{4})$

Step 1.2

Cancel the common factor of $y^{4}$ .

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Step 1.2.1

Factor $y^{4}$ out of $t^{\frac{5}{2}} y^{4}$ .

$\frac{1}{y^{4}} \frac{d y}{d t} = \frac{1}{y^{4}} (y^{4} t^{\frac{5}{2}})$

Step 1.2.2

Cancel the common factor.

$\frac{1}{y^{4}} \frac{d y}{d t} = \frac{1}{y^{4}} (y^{4} t^{\frac{5}{2}})$

Step 1.2.3

Rewrite the expression.

$\frac{1}{y^{4}} \frac{d y}{d t} = t^{\frac{5}{2}}$

Step 1.3

Rewrite the equation.

$\frac{1}{y^{4}} d y = t^{\frac{5}{2}} d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{y^{4}} d y = \int t^{\frac{5}{2}} d t$

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 $y^{4}$ out of the denominator by raising it to the $- 1$ power.

$\int {(y^{4})}^{- 1} d y = \int t^{\frac{5}{2}} d t$

Step 2.2.1.2

Multiply the exponents in ${(y^{4})}^{- 1}$ .

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Step 2.2.1.2.1

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

$\int y^{4 \cdot - 1} d y = \int t^{\frac{5}{2}} d t$

Step 2.2.1.2.2

Multiply $4$ by $- 1$ .

$\int y^{- 4} d y = \int t^{\frac{5}{2}} d t$

Step 2.2.2

By the Power Rule, the integral of $y^{- 4}$ with respect to $y$ is $- \frac{1}{3} y^{- 3}$ .

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

Step 2.2.3

Simplify the answer.

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Step 2.2.3.1

Rewrite $- \frac{1}{3} y^{- 3} + C_{1}$ as $- \frac{1}{3} \cdot \frac{1}{y^{3}} + C_{1}$ .

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

Step 2.2.3.2

Simplify.

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Step 2.2.3.2.1

Multiply $\frac{1}{y^{3}}$ by $\frac{1}{3}$ .

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

Step 2.2.3.2.2

Move $3$ to the left of $y^{3}$ .

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

Step 2.3

By the Power Rule, the integral of $t^{\frac{5}{2}}$ with respect to $t$ is $\frac{2}{7} t^{\frac{7}{2}}$ .

$- \frac{1}{3 y^{3}} + C_{1} = \frac{2}{7} t^{\frac{7}{2}} + C_{2}$

Step 2.4

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

$- \frac{1}{3 y^{3}} = \frac{2}{7} t^{\frac{7}{2}} + K$