Calculus Examples

$\frac{d y}{d x} = e^{- x^{2}}$ , $y (3) = 5$

Step 1

Rewrite the equation.

$d y = e^{- x^{2}} d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int d y = \int e^{- x^{2}} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int e^{- x^{2}} d x$

Step 2.3

$\int e^{- x^{2}} d x$ is a special integral. The integral is the error function.

$y + C_{1} = erf (x) + C_{2}$

Step 2.4

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

$y = erf (x) + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $3$ for $x$ and $5$ for $y$ in $y = erf (x) + K$ .

$5 = erf (x) + K$

Step 4

Solve for $K$ .

Tap for more steps...

Step 4.1

Rewrite the equation as $e r f (x) + K = 5$ .

$e r f (x) + K = 5$

Step 4.2

Multiply $e r f$ by $x$ .

$e r f x + K = 5$

Step 4.3

Subtract $e r f x$ from both sides of the equation.

$K = 5 - e r f x$

Step 5

Substitute $5 - e r f x$ for $K$ in $y = erf (x) + K$ and simplify.

Tap for more steps...

Step 5.1

Substitute $5 - e r f x$ for $K$ .

$y = erf (x) + 5 - e r f x$