Calculus Examples

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

Step 1

Let $v = \frac{d y}{d x}$ . Then $\frac{d v}{d x} = \frac{d^{2} y}{d x^{2}}$ . Substitute $v$ for $\frac{d y}{d x}$ and $\frac{d v}{d x}$ for $\frac{d^{2} y}{d x^{2}}$ to get a differential equation with dependent variable $v$ and independent variable $x$ .

$\frac{d v}{d x} = v$

Step 2

Separate the variables.

Tap for more steps...

Step 2.1

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

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

Step 2.2

Cancel the common factor of $v$ .

Tap for more steps...

Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Rewrite the expression.

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

Step 2.3

Rewrite the equation.

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

Step 3

Integrate both sides.

Tap for more steps...

Step 3.1

Set up an integral on each side.

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

Step 3.2

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

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

Step 3.3

Apply the constant rule.

$\ln (| v |) + C_{1} = x + C_{2}$

Step 3.4

Group the constant of integration on the right side as $C_{3}$ .

$\ln (| v |) = x + C_{3}$

Step 4

Solve for $v$ .

Tap for more steps...

Step 4.1

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

$e^{\ln (| v |)} = e^{x + C_{3}}$

Step 4.2

Rewrite $\ln (| v |) = x + C_{3}$ 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 $b^{y} = x$ .

$e^{x + C_{3}} = | v |$

Step 4.3

Solve for $v$ .

Tap for more steps...

Step 4.3.1

Rewrite the equation as $| v | = e^{x + C_{3}}$ .

$| v | = e^{x + C_{3}}$

Step 4.3.2

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

$v = \pm e^{x + C_{3}}$

Step 5

Group the constant terms together.

Tap for more steps...

Step 5.1

Rewrite $e^{x + C_{3}}$ as $e^{x} e^{C_{3}}$ .

$v = \pm e^{x} e^{C_{3}}$

Step 5.2

Reorder $e^{x}$ and $e^{C_{3}}$ .

$v = \pm e^{C_{3}} e^{x}$

Step 5.3

Combine constants with the plus or minus.

$v = C_{4} e^{x}$

Step 6

Replace all occurrences of $v$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = C_{4} e^{x}$

Step 7

Rewrite the equation.

$d y = C_{4} e^{x} d x$

Step 8

Integrate both sides.

Tap for more steps...

Step 8.1

Set up an integral on each side.

$\int d y = \int C_{4} e^{x} d x$

Step 8.2

Apply the constant rule.

$y + C_{5} = \int C_{4} e^{x} d x$

Step 8.3

Integrate the right side.

Tap for more steps...

Step 8.3.1

Since $C_{4}$ is constant with respect to $x$ , move $C_{4}$ out of the integral.

$y + C_{5} = C_{4} \int e^{x} d x$

Step 8.3.2

The integral of $e^{x}$ with respect to $x$ is $e^{x}$ .

$y + C_{5} = C_{4} (e^{x} + C_{6})$

Step 8.3.3

Simplify.

$y + C_{5} = C_{4} e^{x} + C_{7}$

Step 8.3.4

Reorder terms.

$y + C_{5} = e^{x} C_{4} + C_{7}$

Step 8.4

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

$y = e^{x} C + D$