Calculus Examples

$\frac{d y}{d x} = x^{3} (y - 3)$ , $y (0) = 6$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{y - 3}$ .

$\frac{1}{y - 3} \frac{d y}{d x} = \frac{1}{y - 3} (x^{3} (y - 3))$

Step 1.2

Cancel the common factor of $y - 3$ .

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Step 1.2.1

Factor $y - 3$ out of $x^{3} (y - 3)$ .

$\frac{1}{y - 3} \frac{d y}{d x} = \frac{1}{y - 3} ((y - 3) x^{3})$

Step 1.2.2

Cancel the common factor.

$\frac{1}{y - 3} \frac{d y}{d x} = \frac{1}{y - 3} ((y - 3) x^{3})$

Step 1.2.3

Rewrite the expression.

$\frac{1}{y - 3} \frac{d y}{d x} = x^{3}$

Step 1.3

Rewrite the equation.

$\frac{1}{y - 3} d y = x^{3} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{y - 3} d y = \int x^{3} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Let $u = y - 3$ . Then $d u = d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.1.1

Let $u = y - 3$ . Find $\frac{d u}{d y}$ .

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Step 2.2.1.1.1

Differentiate $y - 3$ .

$\frac{d}{d y} [y - 3]$

Step 2.2.1.1.2

By the Sum Rule, the derivative of $y - 3$ with respect to $y$ is $\frac{d}{d y} [y] + \frac{d}{d y} [- 3]$ .

$\frac{d}{d y} [y] + \frac{d}{d y} [- 3]$

Step 2.2.1.1.3

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

$1 + \frac{d}{d y} [- 3]$

Step 2.2.1.1.4

Since $- 3$ is constant with respect to $y$ , the derivative of $- 3$ with respect to $y$ is $0$ .

$1 + 0$

Step 2.2.1.1.5

Add $1$ and $0$ .

$1$

Step 2.2.1.2

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

$\int \frac{1}{u} d u = \int x^{3} d x$

Step 2.2.2

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

$\ln (| u |) + C_{1} = \int x^{3} d x$

Step 2.2.3

Replace all occurrences of $u$ with $y - 3$ .

$\ln (| y - 3 |) + C_{1} = \int x^{3} d x$

Step 2.3

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

$\ln (| y - 3 |) + C_{1} = \frac{1}{4} x^{4} + C_{2}$

Step 2.4

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

$\ln (| y - 3 |) = \frac{1}{4} x^{4} + C$

Step 3

Solve for $y$ .

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Step 3.1

To solve for $y$ , rewrite the equation using properties of logarithms.

$e^{\ln (| y - 3 |)} = e^{\frac{1}{4} x^{4} + C}$

Step 3.2

Rewrite $\ln (| y - 3 |) = \frac{1}{4} x^{4} + C$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{\frac{1}{4} x^{4} + C} = | y - 3 |$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Rewrite the equation as $| y - 3 | = e^{\frac{1}{4} x^{4} + C}$ .

$| y - 3 | = e^{\frac{1}{4} x^{4} + C}$

Step 3.3.2

Combine $\frac{1}{4}$ and $x^{4}$ .

$| y - 3 | = e^{\frac{x^{4}}{4} + C}$

Step 3.3.3

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$y - 3 = \pm e^{\frac{x^{4}}{4} + C}$

Step 3.3.4

Add $3$ to both sides of the equation.

$y = \pm e^{\frac{x^{4}}{4} + C} + 3$

Step 4

Group the constant terms together.

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Step 4.1

Rewrite $e^{\frac{x^{4}}{4} + C}$ as $e^{\frac{x^{4}}{4}} e^{C}$ .

$y = \pm e^{\frac{x^{4}}{4}} e^{C} + 3$

Step 4.2

Reorder $e^{\frac{x^{4}}{4}}$ and $e^{C}$ .

$y = \pm e^{C} e^{\frac{x^{4}}{4}} + 3$

Step 4.3

Combine constants with the plus or minus.

$y = C e^{\frac{x^{4}}{4}} + 3$

Step 5

Use the initial condition to find the value of $C$ by substituting $0$ for $x$ and $6$ for $y$ in $y = C e^{\frac{x^{4}}{4}} + 3$ .

$6 = C e^{\frac{0^{4}}{4}} + 3$

Step 6

Solve for $C$ .

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Step 6.1

Rewrite the equation as $C e^{\frac{0^{4}}{4}} + 3 = 6$ .

$C e^{\frac{0^{4}}{4}} + 3 = 6$

Step 6.2

Simplify each term.

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Step 6.2.1

Raising $0$ to any positive power yields $0$ .

$C e^{\frac{0}{4}} + 3 = 6$

Step 6.2.2

Divide $0$ by $4$ .

$C e^{0} + 3 = 6$

Step 6.2.3

Anything raised to $0$ is $1$ .

$C \cdot 1 + 3 = 6$

Step 6.2.4

Multiply $C$ by $1$ .

$C + 3 = 6$

Step 6.3

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

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Step 6.3.1

Subtract $3$ from both sides of the equation.

$C = 6 - 3$

Step 6.3.2

Subtract $3$ from $6$ .

$C = 3$

Step 7

Substitute $3$ for $C$ in $y = C e^{\frac{x^{4}}{4}} + 3$ and simplify.

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Step 7.1

Substitute $3$ for $C$ .

$y = 3 e^{\frac{x^{4}}{4}} + 3$