Calculus Examples

$\frac{d y}{d x} = (1 + y^{2}) \tan (x)$ , $y (0) = \sqrt{3}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{1 + y^{2}}$ .

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

Step 1.2

Cancel the common factor of $1 + y^{2}$ .

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Step 1.2.1

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

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

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.

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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.

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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 $C$ .

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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.

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Step 5.3.1

Simplify $| \sec (0) |$ .

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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.

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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 $C$ .

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Step 5.8.1

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

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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.

$No solution$