Calculus Examples

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

Step 1

Assume all solutions are of the form $x = e^{r y}$ .

Step 2

Find the characteristic equation for $y \frac{d^{2} x}{d y^{2}} = y^{2} + 1$ .

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Step 2.1

Find the first derivative.

$\frac{d x}{d y} = r e^{r y}$

Step 2.2

Find the second derivative.

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

Step 2.3

Substitute into the differential equation.

$y (r^{2} e^{r y}) = y^{2} + 1$

Step 2.4

Reorder factors in $y (r^{2} e^{r y})$ .

$y r^{2} e^{r y} = y^{2} + 1$

Step 2.5

Factor $e^{r y}$ out of $y r^{2} e^{r y}$ .

$e^{r y} (y r^{2}) = y^{2} + 1$

Step 2.6

Since exponentials can never be zero, divide both sides by $e^{r y}$ .

$y r^{2} = y^{2} + 1$

Step 3

Solve for $r$ .

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Step 3.1

Divide each term in $y r^{2} = y^{2} + 1$ by $y$ and simplify.

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Step 3.1.1

Divide each term in $y r^{2} = y^{2} + 1$ by $y$ .

$\frac{y r^{2}}{y} = \frac{y^{2}}{y} + \frac{1}{y}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $y$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{y r^{2}}{y} = \frac{y^{2}}{y} + \frac{1}{y}$

Step 3.1.2.1.2

Divide $r^{2}$ by $1$ .

$r^{2} = \frac{y^{2}}{y} + \frac{1}{y}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

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

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Step 3.1.3.1.1

Factor $y$ out of $y^{2}$ .

$r^{2} = \frac{y \cdot y}{y} + \frac{1}{y}$

Step 3.1.3.1.2

Cancel the common factors.

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Step 3.1.3.1.2.1

Raise $y$ to the power of $1$ .

$r^{2} = \frac{y \cdot y}{y^{1}} + \frac{1}{y}$

Step 3.1.3.1.2.2

Factor $y$ out of $y^{1}$ .

$r^{2} = \frac{y \cdot y}{y \cdot 1} + \frac{1}{y}$

Step 3.1.3.1.2.3

Cancel the common factor.

$r^{2} = \frac{y \cdot y}{y \cdot 1} + \frac{1}{y}$

Step 3.1.3.1.2.4

Rewrite the expression.

$r^{2} = \frac{y}{1} + \frac{1}{y}$

Step 3.1.3.1.2.5

Divide $y$ by $1$ .

$r^{2} = y + \frac{1}{y}$

Step 3.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$r = \pm \sqrt{y + \frac{1}{y}}$

Step 3.3

Simplify $\pm \sqrt{y + \frac{1}{y}}$ .

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Step 3.3.1

To write $y$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$r = \pm \sqrt{\frac{y \cdot y}{y} + \frac{1}{y}}$

Step 3.3.2

Combine the numerators over the common denominator.

$r = \pm \sqrt{\frac{y \cdot y + 1}{y}}$

Step 3.3.3

Multiply $y$ by $y$ .

$r = \pm \sqrt{\frac{y^{2} + 1}{y}}$

Step 3.3.4

Rewrite $\sqrt{\frac{y^{2} + 1}{y}}$ as $\frac{\sqrt{y^{2} + 1}}{\sqrt{y}}$ .

$r = \pm \frac{\sqrt{y^{2} + 1}}{\sqrt{y}}$

Step 3.3.5

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

$r = \pm \frac{\sqrt{y^{2} + 1}}{\sqrt{y}} \cdot \frac{\sqrt{y}}{\sqrt{y}}$

Step 3.3.6

Combine and simplify the denominator.

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Step 3.3.6.1

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

$r = \pm \frac{\sqrt{y^{2} + 1} \sqrt{y}}{\sqrt{y} \sqrt{y}}$

Step 3.3.6.2

Raise $\sqrt{y}$ to the power of $1$ .

$r = \pm \frac{\sqrt{y^{2} + 1} \sqrt{y}}{{\sqrt{y}}^{1} \sqrt{y}}$

Step 3.3.6.3

Raise $\sqrt{y}$ to the power of $1$ .

$r = \pm \frac{\sqrt{y^{2} + 1} \sqrt{y}}{{\sqrt{y}}^{1} {\sqrt{y}}^{1}}$

Step 3.3.6.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$r = \pm \frac{\sqrt{y^{2} + 1} \sqrt{y}}{{\sqrt{y}}^{1 + 1}}$

Step 3.3.6.5

Add $1$ and $1$ .

$r = \pm \frac{\sqrt{y^{2} + 1} \sqrt{y}}{{\sqrt{y}}^{2}}$

Step 3.3.6.6

Rewrite ${\sqrt{y}}^{2}$ as $y$ .

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Step 3.3.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y}$ as $y^{\frac{1}{2}}$ .

$r = \pm \frac{\sqrt{y^{2} + 1} \sqrt{y}}{{(y^{\frac{1}{2}})}^{2}}$

Step 3.3.6.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$r = \pm \frac{\sqrt{y^{2} + 1} \sqrt{y}}{y^{\frac{1}{2} \cdot 2}}$

Step 3.3.6.6.3

Combine $\frac{1}{2}$ and $2$ .

$r = \pm \frac{\sqrt{y^{2} + 1} \sqrt{y}}{y^{\frac{2}{2}}}$

Step 3.3.6.6.4

Cancel the common factor of $2$ .

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Step 3.3.6.6.4.1

Cancel the common factor.

$r = \pm \frac{\sqrt{y^{2} + 1} \sqrt{y}}{y^{\frac{2}{2}}}$

Step 3.3.6.6.4.2

Rewrite the expression.

$r = \pm \frac{\sqrt{y^{2} + 1} \sqrt{y}}{y^{1}}$

Step 3.3.6.6.5

Simplify.

$r = \pm \frac{\sqrt{y^{2} + 1} \sqrt{y}}{y}$

Step 3.3.7

Combine using the product rule for radicals.

$r = \pm \frac{\sqrt{(y^{2} + 1) y}}{y}$

Step 3.3.8

Reorder factors in $\pm \frac{\sqrt{(y^{2} + 1) y}}{y}$ .

$r = \pm \frac{\sqrt{y (y^{2} + 1)}}{y}$

Step 3.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.4.1

First, use the positive value of the $\pm$ to find the first solution.

$r = \frac{\sqrt{y (y^{2} + 1)}}{y}$

Step 3.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$r = - \frac{\sqrt{y (y^{2} + 1)}}{y}$

Step 3.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$r = \frac{\sqrt{y (y^{2} + 1)}}{y}$

$r = - \frac{\sqrt{y (y^{2} + 1)}}{y}$

$r = \frac{\sqrt{y (y^{2} + 1)}}{y}$

$r = - \frac{\sqrt{y (y^{2} + 1)}}{y}$

$r = \frac{\sqrt{y (y^{2} + 1)}}{y}$

$r = - \frac{\sqrt{y (y^{2} + 1)}}{y}$

Step 4

With the two found values of $r$ , two solutions can be constructed.

$x_{1} = e^{\frac{\sqrt{y (y^{2} + 1)}}{y} y}$

$x_{2} = e^{- \frac{\sqrt{y (y^{2} + 1)}}{y} y}$

Step 5

By the principle of superposition, the general solution is a linear combination of the two solutions for a second order homogeneous linear differential equation.

$x = c_{1} e^{\frac{\sqrt{y (y^{2} + 1)}}{y} y} + c_{2} e^{- \frac{\sqrt{y (y^{2} + 1)}}{y} y}$

Step 6

Simplify each term.

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Step 6.1

Cancel the common factor of $y$ .

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Step 6.1.1

Cancel the common factor.

$x = c_{1} e^{\frac{\sqrt{y (y^{2} + 1)}}{y} y} + c_{2} e^{- \frac{\sqrt{y (y^{2} + 1)}}{y} y}$

Step 6.1.2

Rewrite the expression.

$x = c_{1} e^{\sqrt{y (y^{2} + 1)}} + c_{2} e^{- \frac{\sqrt{y (y^{2} + 1)}}{y} y}$

Step 6.2

Cancel the common factor of $y$ .

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Step 6.2.1

Move the leading negative in $- \frac{\sqrt{y (y^{2} + 1)}}{y}$ into the numerator.

$x = c_{1} e^{\sqrt{y (y^{2} + 1)}} + c_{2} e^{\frac{- \sqrt{y (y^{2} + 1)}}{y} y}$

Step 6.2.2

Cancel the common factor.

$x = c_{1} e^{\sqrt{y (y^{2} + 1)}} + c_{2} e^{\frac{- \sqrt{y (y^{2} + 1)}}{y} y}$

Step 6.2.3

Rewrite the expression.

$x = c_{1} e^{\sqrt{y (y^{2} + 1)}} + c_{2} e^{- \sqrt{y (y^{2} + 1)}}$