Calculus Examples

$\frac{d y}{d x} = x y$ , $y (0) = 1$

Step 1

Separate the variables.

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Step 1.1

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

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

Step 1.2

Cancel the common factor of $y$ .

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Step 1.2.1

Factor $y$ out of $x y$ .

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

Step 1.2.2

Cancel the common factor.

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

Step 1.2.3

Rewrite the expression.

$\frac{1}{y} \frac{d y}{d x} = x$

Step 1.3

Rewrite the equation.

$\frac{1}{y} d y = x d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{y} d y = \int x d x$

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

$\ln (| y |) = \frac{1}{2} x^{2} + C$

Step 3

Solve for $y$ .

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Step 3.1

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

$e^{\ln (| y |)} = e^{\frac{1}{2} x^{2} + C}$

Step 3.2

Rewrite $\ln (| y |) = \frac{1}{2} x^{2} + C$ 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 $b^{y} = x$ .

$e^{\frac{1}{2} x^{2} + C} = | y |$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Rewrite the equation as $| y | = e^{\frac{1}{2} x^{2} + C}$ .

$| y | = e^{\frac{1}{2} x^{2} + C}$

Step 3.3.2

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

$| y | = e^{\frac{x^{2}}{2} + C}$

Step 3.3.3

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

$y = \pm e^{\frac{x^{2}}{2} + C}$

Step 4

Group the constant terms together.

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Step 4.1

Rewrite $e^{\frac{x^{2}}{2} + C}$ as $e^{\frac{x^{2}}{2}} e^{C}$ .

$y = \pm e^{\frac{x^{2}}{2}} e^{C}$

Step 4.2

Reorder $e^{\frac{x^{2}}{2}}$ and $e^{C}$ .

$y = \pm e^{C} e^{\frac{x^{2}}{2}}$

Step 4.3

Combine constants with the plus or minus.

$y = C e^{\frac{x^{2}}{2}}$

Step 5

Use the initial condition to find the value of $C$ by substituting $0$ for $x$ and $1$ for $y$ in $y = C e^{\frac{x^{2}}{2}}$ .

$1 = C e^{\frac{0^{2}}{2}}$

Step 6

Solve for $C$ .

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Step 6.1

Rewrite the equation as $C e^{\frac{0^{2}}{2}} = 1$ .

$C e^{\frac{0^{2}}{2}} = 1$

Step 6.2

Simplify $C e^{\frac{0^{2}}{2}}$ .

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Step 6.2.1

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

$C e^{\frac{0}{2}} = 1$

Step 6.2.2

Divide $0$ by $2$ .

$C e^{0} = 1$

Step 6.2.3

Anything raised to $0$ is $1$ .

$C \cdot 1 = 1$

Step 6.2.4

Multiply $C$ by $1$ .

$C = 1$

Step 7

Substitute $1$ for $C$ in $y = C e^{\frac{x^{2}}{2}}$ and simplify.

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Step 7.1

Substitute $1$ for $C$ .

$y = 1 e^{\frac{x^{2}}{2}}$

Step 7.2

Multiply $e^{\frac{x^{2}}{2}}$ by $1$ .

$y = e^{\frac{x^{2}}{2}}$