Calculus Examples

\frac{d y}{d x} = 13 x y

$\frac{d y}{d x} = 13 x y$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by

\frac{1}{y}

$\frac{1}{y}$ .

\frac{1}{y} \frac{d y}{d x} = \frac{1}{y} (13 x y)

$\frac{1}{y} \frac{d y}{d x} = \frac{1}{y} (13 x y)$

Step 1.2

Simplify.

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Step 1.2.1

Rewrite using the commutative property of multiplication.

\frac{1}{y} \frac{d y}{d x} = 13 \frac{1}{y} (x y)

$\frac{1}{y} \frac{d y}{d x} = 13 \frac{1}{y} (x y)$

Step 1.2.2

Combine

13

$13$ and

\frac{1}{y}

$\frac{1}{y}$ .

\frac{1}{y} \frac{d y}{d x} = \frac{13}{y} (x y)

$\frac{1}{y} \frac{d y}{d x} = \frac{13}{y} (x y)$

Step 1.2.3

Cancel the common factor of

y

$y$ .

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Step 1.2.3.1

Factor

y

$y$ out of

x y

$x y$ .

\frac{1}{y} \frac{d y}{d x} = \frac{13}{y} (y x)

$\frac{1}{y} \frac{d y}{d x} = \frac{13}{y} (y x)$

Step 1.2.3.2

Cancel the common factor.

$\frac{1}{y} \frac{d y}{d x} = \frac{13}{y} (y x)$

Step 1.2.3.3

Rewrite the expression.

$\frac{1}{y} \frac{d y}{d x} = 13 x$

Step 1.3

Rewrite the equation.

$\frac{1}{y} d y = 13 x d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{y} d y = \int 13 x d x$

Step 2.2

The integral of

$\frac{1}{y}$ with respect to

$y$ is

$\ln (| y |)$ .

$\ln (| y |) + C_{1} = \int 13 x d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since

$13$ is constant with respect to

$x$ , move

$13$ out of the integral.

$\ln (| y |) + C_{1} = 13 \int x d x$

Step 2.3.2

By the Power Rule, the integral of

$x$ with respect to

$x$ is

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

$\ln (| y |) + C_{1} = 13 (\frac{1}{2} x^{2} + C_{2})$

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite

$13 (\frac{1}{2} x^{2} + C_{2})$ as

$13 (\frac{1}{2}) x^{2} + C_{2}$ .

$\ln (| y |) + C_{1} = 13 (\frac{1}{2}) x^{2} + C_{2}$

Step 2.3.3.2

Combine

$13$ and

$\frac{1}{2}$ .

$\ln (| y |) + C_{1} = \frac{13}{2} x^{2} + C_{2}$

Step 2.4

Group the constant of integration on the right side as

$C$ .

$\ln (| y |) = \frac{13}{2} x^{2} + C$

Step 3

Solve for

$y$ .

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Step 3.1

To solve for

$y$ , rewrite the equation using properties of logarithms.

$e^{\ln (| y |)} = e^{\frac{13}{2} x^{2} + C}$

Step 3.2

Rewrite

$\ln (| y |) = \frac{13}{2} x^{2} + C$ 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

$b^{y} = x$ .

$e^{\frac{13}{2} x^{2} + C} = | y |$

Step 3.3

Solve for

$y$ .

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Step 3.3.1

Rewrite the equation as

$| y | = e^{\frac{13}{2} x^{2} + C}$ .

$| y | = e^{\frac{13}{2} x^{2} + C}$

Step 3.3.2

Combine

$\frac{13}{2}$ and

$x^{2}$ .

$| y | = e^{\frac{13 x^{2}}{2} + C}$

Step 3.3.3

Remove the absolute value term. This creates a

$\pm$ on the right side of the equation because

$| x | = \pm x$ .

$y = \pm e^{\frac{13 x^{2}}{2} + C}$

Step 4

Group the constant terms together.

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Step 4.1

Rewrite

$e^{\frac{13 x^{2}}{2} + C}$ as

$e^{\frac{13 x^{2}}{2}} e^{C}$ .

$y = \pm e^{\frac{13 x^{2}}{2}} e^{C}$

Step 4.2

Reorder

$e^{\frac{13 x^{2}}{2}}$ and

$e^{C}$ .

$y = \pm e^{C} e^{\frac{13 x^{2}}{2}}$

Step 4.3

Combine constants with the plus or minus.

$y = C e^{\frac{13 x^{2}}{2}}$