Calculus Examples

$\frac{d y}{d t} = 2 - \frac{3 y}{25 + 2 t}$

Step 1

Rewrite the differential equation as $\frac{d y}{d t} + M (t) y = Q (t)$ .

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Step 1.1

Rewrite the equation as $M (t) \frac{d y}{d t} + P (t) y = Q (t)$ .

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Step 1.1.1

Add $\frac{3 y}{25 + 2 t}$ to both sides of the equation.

$\frac{d y}{d t} + \frac{3 y}{25 + 2 t} = 2$

Step 1.1.2

Reorder terms.

$\frac{d y}{d t} + \frac{3 y}{2 t + 25} = 2$

Step 1.2

Factor $y$ out of $\frac{3 y}{2 t + 25}$ .

$\frac{d y}{d t} + y \frac{3}{2 t + 25} = 2$

Step 1.3

Reorder $y$ and $\frac{3}{2 t + 25}$ .

$\frac{d y}{d t} + \frac{3}{2 t + 25} y = 2$

Step 2

The integrating factor is defined by the formula $e^{\int P (t) d t}$ , where $P (t) = \frac{3}{2 t + 25}$ .

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Step 2.1

Set up the integration.

$e^{\int \frac{3}{2 t + 25} d t}$

Step 2.2

Integrate $\frac{3}{2 t + 25}$ .

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Step 2.2.1

Since $3$ is constant with respect to $t$ , move $3$ out of the integral.

$e^{3 \int \frac{1}{2 t + 25} d t}$

Step 2.2.2

Let $u = 2 t + 25$ . Then $d u = 2 d t$ , so $\frac{1}{2} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.2.1

Let $u = 2 t + 25$ . Find $\frac{d u}{d t}$ .

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Step 2.2.2.1.1

Differentiate $2 t + 25$ .

$\frac{d}{d t} [2 t + 25]$

Step 2.2.2.1.2

By the Sum Rule, the derivative of $2 t + 25$ with respect to $t$ is $\frac{d}{d t} [2 t] + \frac{d}{d t} [25]$ .

$\frac{d}{d t} [2 t] + \frac{d}{d t} [25]$

Step 2.2.2.1.3

Evaluate $\frac{d}{d t} [2 t]$ .

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Step 2.2.2.1.3.1

Since $2$ is constant with respect to $t$ , the derivative of $2 t$ with respect to $t$ is $2 \frac{d}{d t} [t]$ .

$2 \frac{d}{d t} [t] + \frac{d}{d t} [25]$

Step 2.2.2.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1$ .

$2 \cdot 1 + \frac{d}{d t} [25]$

Step 2.2.2.1.3.3

Multiply $2$ by $1$ .

$2 + \frac{d}{d t} [25]$

Step 2.2.2.1.4

Differentiate using the Constant Rule.

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Step 2.2.2.1.4.1

Since $25$ is constant with respect to $t$ , the derivative of $25$ with respect to $t$ is $0$ .

$2 + 0$

Step 2.2.2.1.4.2

Add $2$ and $0$ .

$2$

Step 2.2.2.2

Rewrite the problem using $u$ and $d u$ .

$e^{3 \int \frac{1}{u} \cdot \frac{1}{2} d u}$

Step 2.2.3

Simplify.

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Step 2.2.3.1

Multiply $\frac{1}{u}$ by $\frac{1}{2}$ .

$e^{3 \int \frac{1}{u \cdot 2} d u}$

Step 2.2.3.2

Move $2$ to the left of $u$ .

$e^{3 \int \frac{1}{2 u} d u}$

Step 2.2.4

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$e^{3 (\frac{1}{2} \int \frac{1}{u} d u)}$

Step 2.2.5

Combine $\frac{1}{2}$ and $3$ .

$e^{\frac{3}{2} \int \frac{1}{u} d u}$

Step 2.2.6

The integral of $\frac{1}{u}$ with respect to $u$ is $\ln (| u |)$ .

$e^{\frac{3}{2} (\ln (| u |) + C)}$

Step 2.2.7

Simplify.

$e^{\frac{3}{2} \ln (| u |) + C}$

Step 2.2.8

Replace all occurrences of $u$ with $2 t + 25$ .

$e^{\frac{3}{2} \ln (| 2 t + 25 |) + C}$

Step 2.3

Remove the constant of integration.

$e^{\frac{3}{2} \ln (2 t + 25)}$

Step 2.4

Use the logarithmic power rule.

$e^{\ln ({(2 t + 25)}^{\frac{3}{2}})}$

Step 2.5

Exponentiation and log are inverse functions.

${(2 t + 25)}^{\frac{3}{2}}$

Step 3

Multiply each term by the integrating factor ${(2 t + 25)}^{\frac{3}{2}}$ .

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Step 3.1

Multiply each term by ${(2 t + 25)}^{\frac{3}{2}}$ .

${(2 t + 25)}^{\frac{3}{2}} \frac{d y}{d t} + {(2 t + 25)}^{\frac{3}{2}} (\frac{3}{2 t + 25} y) = {(2 t + 25)}^{\frac{3}{2}} \cdot 2$

Step 3.2

Simplify each term.

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Step 3.2.1

Combine $\frac{3}{2 t + 25}$ and $y$ .

${(2 t + 25)}^{\frac{3}{2}} \frac{d y}{d t} + {(2 t + 25)}^{\frac{3}{2}} \frac{3 y}{2 t + 25} = {(2 t + 25)}^{\frac{3}{2}} \cdot 2$

Step 3.2.2

Combine ${(2 t + 25)}^{\frac{3}{2}}$ and $\frac{3 y}{2 t + 25}$ .

${(2 t + 25)}^{\frac{3}{2}} \frac{d y}{d t} + \frac{{(2 t + 25)}^{\frac{3}{2}} (3 y)}{2 t + 25} = {(2 t + 25)}^{\frac{3}{2}} \cdot 2$

Step 3.2.3

Factor $2 t + 25$ out of ${(2 t + 25)}^{\frac{3}{2}} (3 y)$ .

${(2 t + 25)}^{\frac{3}{2}} \frac{d y}{d t} + \frac{(2 t + 25) ({(2 t + 25)}^{\frac{1}{2}} (3 y))}{2 t + 25} = {(2 t + 25)}^{\frac{3}{2}} \cdot 2$

Step 3.2.4

Cancel the common factors.

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Step 3.2.4.1

Multiply by $1$ .

${(2 t + 25)}^{\frac{3}{2}} \frac{d y}{d t} + \frac{(2 t + 25) ({(2 t + 25)}^{\frac{1}{2}} (3 y))}{(2 t + 25) \cdot 1} = {(2 t + 25)}^{\frac{3}{2}} \cdot 2$

Step 3.2.4.2

Cancel the common factor.

${(2 t + 25)}^{\frac{3}{2}} \frac{d y}{d t} + \frac{(2 t + 25) ({(2 t + 25)}^{\frac{1}{2}} (3 y))}{(2 t + 25) \cdot 1} = {(2 t + 25)}^{\frac{3}{2}} \cdot 2$

Step 3.2.4.3

Rewrite the expression.

${(2 t + 25)}^{\frac{3}{2}} \frac{d y}{d t} + \frac{{(2 t + 25)}^{\frac{1}{2}} (3 y)}{1} = {(2 t + 25)}^{\frac{3}{2}} \cdot 2$

Step 3.2.4.4

Divide ${(2 t + 25)}^{\frac{1}{2}} (3 y)$ by $1$ .

${(2 t + 25)}^{\frac{3}{2}} \frac{d y}{d t} + {(2 t + 25)}^{\frac{1}{2}} (3 y) = {(2 t + 25)}^{\frac{3}{2}} \cdot 2$

Step 3.2.5

Rewrite using the commutative property of multiplication.

${(2 t + 25)}^{\frac{3}{2}} \frac{d y}{d t} + 3 {(2 t + 25)}^{\frac{1}{2}} y = {(2 t + 25)}^{\frac{3}{2}} \cdot 2$

Step 3.3

Move $2$ to the left of ${(2 t + 25)}^{\frac{3}{2}}$ .

${(2 t + 25)}^{\frac{3}{2}} \frac{d y}{d t} + 3 {(2 t + 25)}^{\frac{1}{2}} y = 2 \cdot {(2 t + 25)}^{\frac{3}{2}}$

Step 3.4

Reorder factors in ${(2 t + 25)}^{\frac{3}{2}} \frac{d y}{d t} + 3 {(2 t + 25)}^{\frac{1}{2}} y = 2 {(2 t + 25)}^{\frac{3}{2}}$ .

${(2 t + 25)}^{\frac{3}{2}} \frac{d y}{d t} + 3 y {(2 t + 25)}^{\frac{1}{2}} = 2 {(2 t + 25)}^{\frac{3}{2}}$

Step 4

Rewrite the left side as a result of differentiating a product.

$\frac{d}{d t} [{(2 t + 25)}^{\frac{3}{2}} y] = 2 {(2 t + 25)}^{\frac{3}{2}}$

Step 5

Set up an integral on each side.

$\int \frac{d}{d t} [{(2 t + 25)}^{\frac{3}{2}} y] d t = \int 2 {(2 t + 25)}^{\frac{3}{2}} d t$

Step 6

Integrate the left side.

${(2 t + 25)}^{\frac{3}{2}} y = \int 2 {(2 t + 25)}^{\frac{3}{2}} d t$

Step 7

Integrate the right side.

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Step 7.1

Since $2$ is constant with respect to $t$ , move $2$ out of the integral.

${(2 t + 25)}^{\frac{3}{2}} y = 2 \int {(2 t + 25)}^{\frac{3}{2}} d t$

Step 7.2

Let $u = 2 t + 25$ . Then $d u = 2 d t$ , so $\frac{1}{2} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 7.2.1

Let $u = 2 t + 25$ . Find $\frac{d u}{d t}$ .

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Step 7.2.1.1

Differentiate $2 t + 25$ .

$\frac{d}{d t} [2 t + 25]$

Step 7.2.1.2

By the Sum Rule, the derivative of $2 t + 25$ with respect to $t$ is $\frac{d}{d t} [2 t] + \frac{d}{d t} [25]$ .

$\frac{d}{d t} [2 t] + \frac{d}{d t} [25]$

Step 7.2.1.3

Evaluate $\frac{d}{d t} [2 t]$ .

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Step 7.2.1.3.1

Since $2$ is constant with respect to $t$ , the derivative of $2 t$ with respect to $t$ is $2 \frac{d}{d t} [t]$ .

$2 \frac{d}{d t} [t] + \frac{d}{d t} [25]$

Step 7.2.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1$ .

$2 \cdot 1 + \frac{d}{d t} [25]$

Step 7.2.1.3.3

Multiply $2$ by $1$ .

$2 + \frac{d}{d t} [25]$

Step 7.2.1.4

Differentiate using the Constant Rule.

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Step 7.2.1.4.1

Since $25$ is constant with respect to $t$ , the derivative of $25$ with respect to $t$ is $0$ .

$2 + 0$

Step 7.2.1.4.2

Add $2$ and $0$ .

$2$

Step 7.2.2

Rewrite the problem using $u$ and $d u$ .

${(2 t + 25)}^{\frac{3}{2}} y = 2 \int u^{\frac{3}{2}} \frac{1}{2} d u$

Step 7.3

Combine $u^{\frac{3}{2}}$ and $\frac{1}{2}$ .

${(2 t + 25)}^{\frac{3}{2}} y = 2 \int \frac{u^{\frac{3}{2}}}{2} d u$

Step 7.4

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

${(2 t + 25)}^{\frac{3}{2}} y = 2 (\frac{1}{2} \int u^{\frac{3}{2}} d u)$

Step 7.5

Simplify.

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Step 7.5.1

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

${(2 t + 25)}^{\frac{3}{2}} y = \frac{2}{2} \int u^{\frac{3}{2}} d u$

Step 7.5.2

Cancel the common factor of $2$ .

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Step 7.5.2.1

Cancel the common factor.

${(2 t + 25)}^{\frac{3}{2}} y = \frac{2}{2} \int u^{\frac{3}{2}} d u$

Step 7.5.2.2

Rewrite the expression.

${(2 t + 25)}^{\frac{3}{2}} y = 1 \int u^{\frac{3}{2}} d u$

Step 7.5.3

Multiply $\int u^{\frac{3}{2}} d u$ by $1$ .

${(2 t + 25)}^{\frac{3}{2}} y = \int u^{\frac{3}{2}} d u$

Step 7.6

By the Power Rule, the integral of $u^{\frac{3}{2}}$ with respect to $u$ is $\frac{2}{5} u^{\frac{5}{2}}$ .

${(2 t + 25)}^{\frac{3}{2}} y = \frac{2}{5} u^{\frac{5}{2}} + C$

Step 7.7

Replace all occurrences of $u$ with $2 t + 25$ .

${(2 t + 25)}^{\frac{3}{2}} y = \frac{2}{5} {(2 t + 25)}^{\frac{5}{2}} + C$

Step 8

Divide each term in ${(2 t + 25)}^{\frac{3}{2}} y = \frac{2}{5} {(2 t + 25)}^{\frac{5}{2}} + C$ by ${(2 t + 25)}^{\frac{3}{2}}$ and simplify.

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Step 8.1

Divide each term in ${(2 t + 25)}^{\frac{3}{2}} y = \frac{2}{5} {(2 t + 25)}^{\frac{5}{2}} + C$ by ${(2 t + 25)}^{\frac{3}{2}}$ .

$\frac{{(2 t + 25)}^{\frac{3}{2}} y}{{(2 t + 25)}^{\frac{3}{2}}} = \frac{\frac{2}{5} {(2 t + 25)}^{\frac{5}{2}}}{{(2 t + 25)}^{\frac{3}{2}}} + \frac{C}{{(2 t + 25)}^{\frac{3}{2}}}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor.

$\frac{{(2 t + 25)}^{\frac{3}{2}} y}{{(2 t + 25)}^{\frac{3}{2}}} = \frac{\frac{2}{5} {(2 t + 25)}^{\frac{5}{2}}}{{(2 t + 25)}^{\frac{3}{2}}} + \frac{C}{{(2 t + 25)}^{\frac{3}{2}}}$

Step 8.2.2

Divide $y$ by $1$ .

$y = \frac{\frac{2}{5} {(2 t + 25)}^{\frac{5}{2}}}{{(2 t + 25)}^{\frac{3}{2}}} + \frac{C}{{(2 t + 25)}^{\frac{3}{2}}}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Combine the numerators over the common denominator.

$y = \frac{\frac{2}{5} {(2 t + 25)}^{\frac{5}{2}} + C}{{(2 t + 25)}^{\frac{3}{2}}}$

Step 8.3.2

Combine $\frac{2}{5}$ and ${(2 t + 25)}^{\frac{5}{2}}$ .

$y = \frac{\frac{2 {(2 t + 25)}^{\frac{5}{2}}}{5} + C}{{(2 t + 25)}^{\frac{3}{2}}}$

Step 8.3.3

To write $C$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = \frac{\frac{2 {(2 t + 25)}^{\frac{5}{2}}}{5} + C \cdot \frac{5}{5}}{{(2 t + 25)}^{\frac{3}{2}}}$

Step 8.3.4

Simplify terms.

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Step 8.3.4.1

Combine $C$ and $\frac{5}{5}$ .

$y = \frac{\frac{2 {(2 t + 25)}^{\frac{5}{2}}}{5} + \frac{C \cdot 5}{5}}{{(2 t + 25)}^{\frac{3}{2}}}$

Step 8.3.4.2

Combine the numerators over the common denominator.

$y = \frac{\frac{2 {(2 t + 25)}^{\frac{5}{2}} + C \cdot 5}{5}}{{(2 t + 25)}^{\frac{3}{2}}}$

Step 8.3.5

Move $5$ to the left of $C$ .

$y = \frac{\frac{2 {(2 t + 25)}^{\frac{5}{2}} + 5 C}{5}}{{(2 t + 25)}^{\frac{3}{2}}}$

Step 8.3.6

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{2 {(2 t + 25)}^{\frac{5}{2}} + 5 C}{5} \cdot \frac{1}{{(2 t + 25)}^{\frac{3}{2}}}$

Step 8.3.7

Multiply $\frac{2 {(2 t + 25)}^{\frac{5}{2}} + 5 C}{5}$ by $\frac{1}{{(2 t + 25)}^{\frac{3}{2}}}$ .

$y = \frac{2 {(2 t + 25)}^{\frac{5}{2}} + 5 C}{5 {(2 t + 25)}^{\frac{3}{2}}}$