Calculus Examples

\frac{d y}{d x} = (1 + y^{2}) tan (x)

$\frac{d y}{d x} = (1 + y^{2}) \tan (x)$ ,

y (0) = \sqrt{3}

$y (0) = \sqrt{3}$

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

Multiply both sides by

\frac{1}{1 + y^{2}}

$\frac{1}{1 + y^{2}}$ .

\frac{1}{1 + y^{2}} \frac{d y}{d x} = \frac{1}{1 + y^{2}} ((1 + y^{2}) tan (x))

$\frac{1}{1 + y^{2}} \frac{d y}{d x} = \frac{1}{1 + y^{2}} ((1 + y^{2}) \tan (x))$

Step 1.2

Cancel the common factor of

1 + y^{2}

$1 + y^{2}$ .

Tap for more steps...

Step 1.2.1

Factor

1 + y^{2}

$1 + y^{2}$ out of

(1 + y^{2}) tan (x)

$(1 + y^{2}) \tan (x)$ .

\frac{1}{1 + y^{2}} \frac{d y}{d x} = \frac{1}{1 + y^{2}} ((1 + y^{2}) (tan (x)))

$\frac{1}{1 + y^{2}} \frac{d y}{d x} = \frac{1}{1 + y^{2}} ((1 + y^{2}) (\tan (x)))$

Step 1.2.2

Cancel the common factor.

$\frac{1}{1 + y^{2}} \frac{d y}{d x} = \frac{1}{1 + y^{2}} ((1 + y^{2}) \tan (x))$

Step 1.2.3

Rewrite the expression.

$\frac{1}{1 + y^{2}} \frac{d y}{d x} = \tan (x)$

Step 1.3

Rewrite the equation.

$\frac{1}{1 + y^{2}} d y = \tan (x) d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int \frac{1}{1 + y^{2}} d y = \int \tan (x) d x$

Step 2.2

Integrate the left side.

Tap for more steps...

Step 2.2.1

Rewrite

$1$ as

$1^{2}$ .

$\int \frac{1}{1^{2} + y^{2}} d y = \int \tan (x) d x$

Step 2.2.2

The integral of

$\frac{1}{1^{2} + y^{2}}$ with respect to

$y$ is

$\arctan (y) + C_{1}$ .

$\arctan (y) + C_{1} = \int \tan (x) d x$

Step 2.3

The integral of

$\tan (x)$ with respect to

$x$ is

$\ln (| \sec (x) |)$ .

$\arctan (y) + C_{1} = \ln (| \sec (x) |) + C_{2}$

Step 2.4

Group the constant of integration on the right side as

$C$ .

$\arctan (y) = \ln (| \sec (x) |) + C$

Step 3

Take the inverse arctangent of both sides of the equation to extract

$y$ from inside the arctangent.

$y = \tan (\ln (| \sec (x) |) + C)$

Step 4

Use the initial condition to find the value of

$C$ by substituting

$0$ for

$x$ and

$\sqrt{3}$ for

$y$ in

$y = \tan (\ln (| \sec (x) |) + C)$ .

$\sqrt{3} = \tan (\ln (| \sec (0) |) + C)$

Step 5

Solve for

$C$ .

Tap for more steps...

Step 5.1

Rewrite the equation as

$\tan (\ln (| \sec (0) |) + C) = \sqrt{3}$ .

$\tan (\ln (| \sec (0) |) + C) = \sqrt{3}$

Step 5.2

Take the inverse tangent of both sides of the equation to extract

$C$ from inside the tangent.

$| \sec (0) | = \arctan (\sqrt{3})$

Step 5.3

Simplify the left side.

Tap for more steps...

Step 5.3.1

Simplify

$| \sec (0) |$ .

Tap for more steps...

Step 5.3.1.1

The exact value of

$\sec (0)$ is

$1$ .

$| 1 | = \arctan (\sqrt{3})$

Step 5.3.1.2

The absolute value is the distance between a number and zero. The distance between

$0$ and

$1$ is

$1$ .

$1 = \arctan (\sqrt{3})$

Step 5.4

Simplify the right side.

Tap for more steps...

Step 5.4.1

The exact value of

$\arctan (\sqrt{3})$ is

$\frac{π}{3}$ .

$1 = \frac{π}{3}$

Step 5.5

Divide

$3.14159265$ by

$3$ .

$1 = 1.04719755$

Step 5.6

Since

$1 \neq 1.04719755$ , there are no solutions.

No solution

Step 5.7

The tangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from

$π$ to find the solution in the fourth quadrant.

$1 = π + \frac{π}{3}$

Step 5.8

Solve for

$C$ .

Tap for more steps...

Step 5.8.1

Simplify

$(3.14159265) + \frac{3.14159265}{3}$ .

Tap for more steps...

Step 5.8.1.1

Divide

$3.14159265$ by

$3$ .

$1 = 3.14159265 + 1.04719755$

Step 5.8.1.2

Add

$3.14159265$ and

$1.04719755$ .

$1 = 4.1887902$

Step 5.8.2

Since

$1 \neq 4.1887902$ , there are no solutions.

$No solution$