Calculus Examples

$\frac{d^{2} y}{d x^{2}} + 4 \frac{d y}{d x} + 4 y = 2 x + 6$

Step 1

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

Step 2

Find the characteristic equation for $\frac{d^{2} y}{d x^{2}} + 4 \frac{d y}{d x} + 4 y = 2 x + 6$ .

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Step 2.1

Find the first derivative.

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

Step 2.2

Find the second derivative.

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

Step 2.3

Substitute into the differential equation.

$r^{2} e^{r x} + 4 (r e^{r x}) + 4 e^{r x} = 2 x + 6$

Step 2.4

Remove parentheses.

$r^{2} e^{r x} + 4 r e^{r x} + 4 e^{r x} = 2 x + 6$

Step 2.5

Factor out $e^{r x}$ .

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Step 2.5.1

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

$e^{r x} r^{2} + 4 r e^{r x} + 4 e^{r x} = 2 x + 6$

Step 2.5.2

Factor $e^{r x}$ out of $4 r e^{r x}$ .

$e^{r x} r^{2} + e^{r x} (4 r) + 4 e^{r x} = 2 x + 6$

Step 2.5.3

Factor $e^{r x}$ out of $e^{r x} r^{2} + e^{r x} (4 r)$ .

$e^{r x} (r^{2} + 4 r) + 4 e^{r x} = 2 x + 6$

Step 2.5.4

Factor $e^{r x}$ out of $4 e^{r x}$ .

$e^{r x} (r^{2} + 4 r) + e^{r x} \cdot 4 = 2 x + 6$

Step 2.5.5

Factor $e^{r x}$ out of $e^{r x} (r^{2} + 4 r) + e^{r x} \cdot 4$ .

$e^{r x} (r^{2} + 4 r + 4) = 2 x + 6$

Step 2.6

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

$r^{2} + 4 r + 4 = 2 x + 6$

Step 3

Solve for $r$ .

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Step 3.1

Move all terms to the left side of the equation and simplify.

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Step 3.1.1

Move all the expressions to the left side of the equation.

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Step 3.1.1.1

Subtract $2 x$ from both sides of the equation.

$r^{2} + 4 r + 4 - 2 x = 6$

Step 3.1.1.2

Subtract $6$ from both sides of the equation.

$r^{2} + 4 r + 4 - 2 x - 6 = 0$

Step 3.1.2

Subtract $6$ from $4$ .

$r^{2} + 4 r - 2 x - 2 = 0$

Step 3.2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3.3

Substitute the values $a = 1$ , $b = 4$ , and $c = - 2 x - 2$ into the quadratic formula and solve for $r$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (1 \cdot (- 2 x - 2))}}{2 \cdot 1}$

Step 3.4

Simplify.

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Step 3.4.1

Simplify the numerator.

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Step 3.4.1.1

Raise $4$ to the power of $2$ .

$r = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot (- 2 x - 2)}}{2 \cdot 1}$

Step 3.4.1.2

Multiply $- 4$ by $1$ .

$r = \frac{- 4 \pm \sqrt{16 - 4 \cdot (- 2 x - 2)}}{2 \cdot 1}$

Step 3.4.1.3

Apply the distributive property.

$r = \frac{- 4 \pm \sqrt{16 - 4 (- 2 x) - 4 \cdot - 2}}{2 \cdot 1}$

Step 3.4.1.4

Multiply $- 2$ by $- 4$ .

$r = \frac{- 4 \pm \sqrt{16 + 8 x - 4 \cdot - 2}}{2 \cdot 1}$

Step 3.4.1.5

Multiply $- 4$ by $- 2$ .

$r = \frac{- 4 \pm \sqrt{16 + 8 x + 8}}{2 \cdot 1}$

Step 3.4.1.6

Add $16$ and $8$ .

$r = \frac{- 4 \pm \sqrt{8 x + 24}}{2 \cdot 1}$

Step 3.4.1.7

Factor $8$ out of $8 x + 24$ .

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Step 3.4.1.7.1

Factor $8$ out of $8 x$ .

$r = \frac{- 4 \pm \sqrt{8 (x) + 24}}{2 \cdot 1}$

Step 3.4.1.7.2

Factor $8$ out of $24$ .

$r = \frac{- 4 \pm \sqrt{8 x + 8 \cdot 3}}{2 \cdot 1}$

Step 3.4.1.7.3

Factor $8$ out of $8 x + 8 \cdot 3$ .

$r = \frac{- 4 \pm \sqrt{8 (x + 3)}}{2 \cdot 1}$

Step 3.4.1.8

Rewrite $8 (x + 3)$ as $2^{2} \cdot (2 (x + 3))$ .

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Step 3.4.1.8.1

Factor $4$ out of $8$ .

$r = \frac{- 4 \pm \sqrt{4 (2) (x + 3)}}{2 \cdot 1}$

Step 3.4.1.8.2

Rewrite $4$ as $2^{2}$ .

$r = \frac{- 4 \pm \sqrt{2^{2} \cdot (2 (x + 3))}}{2 \cdot 1}$

Step 3.4.1.8.3

Add parentheses.

$r = \frac{- 4 \pm \sqrt{2^{2} \cdot (2 (x + 3))}}{2 \cdot 1}$

Step 3.4.1.9

Pull terms out from under the radical.

$r = \frac{- 4 \pm 2 \sqrt{2 (x + 3)}}{2 \cdot 1}$

Step 3.4.2

Multiply $2$ by $1$ .

$r = \frac{- 4 \pm 2 \sqrt{2 (x + 3)}}{2}$

Step 3.4.3

Simplify $\frac{- 4 \pm 2 \sqrt{2 (x + 3)}}{2}$ .

$r = - 2 \pm \sqrt{2 (x + 3)}$

Step 3.5

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 3.5.1

Simplify the numerator.

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Step 3.5.1.1

Raise $4$ to the power of $2$ .

$r = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot (- 2 x - 2)}}{2 \cdot 1}$

Step 3.5.1.2

Multiply $- 4$ by $1$ .

$r = \frac{- 4 \pm \sqrt{16 - 4 \cdot (- 2 x - 2)}}{2 \cdot 1}$

Step 3.5.1.3

Apply the distributive property.

$r = \frac{- 4 \pm \sqrt{16 - 4 (- 2 x) - 4 \cdot - 2}}{2 \cdot 1}$

Step 3.5.1.4

Multiply $- 2$ by $- 4$ .

$r = \frac{- 4 \pm \sqrt{16 + 8 x - 4 \cdot - 2}}{2 \cdot 1}$

Step 3.5.1.5

Multiply $- 4$ by $- 2$ .

$r = \frac{- 4 \pm \sqrt{16 + 8 x + 8}}{2 \cdot 1}$

Step 3.5.1.6

Add $16$ and $8$ .

$r = \frac{- 4 \pm \sqrt{8 x + 24}}{2 \cdot 1}$

Step 3.5.1.7

Factor $8$ out of $8 x + 24$ .

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Step 3.5.1.7.1

Factor $8$ out of $8 x$ .

$r = \frac{- 4 \pm \sqrt{8 (x) + 24}}{2 \cdot 1}$

Step 3.5.1.7.2

Factor $8$ out of $24$ .

$r = \frac{- 4 \pm \sqrt{8 x + 8 \cdot 3}}{2 \cdot 1}$

Step 3.5.1.7.3

Factor $8$ out of $8 x + 8 \cdot 3$ .

$r = \frac{- 4 \pm \sqrt{8 (x + 3)}}{2 \cdot 1}$

Step 3.5.1.8

Rewrite $8 (x + 3)$ as $2^{2} \cdot (2 (x + 3))$ .

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Step 3.5.1.8.1

Factor $4$ out of $8$ .

$r = \frac{- 4 \pm \sqrt{4 (2) (x + 3)}}{2 \cdot 1}$

Step 3.5.1.8.2

Rewrite $4$ as $2^{2}$ .

$r = \frac{- 4 \pm \sqrt{2^{2} \cdot (2 (x + 3))}}{2 \cdot 1}$

Step 3.5.1.8.3

Add parentheses.

$r = \frac{- 4 \pm \sqrt{2^{2} \cdot (2 (x + 3))}}{2 \cdot 1}$

Step 3.5.1.9

Pull terms out from under the radical.

$r = \frac{- 4 \pm 2 \sqrt{2 (x + 3)}}{2 \cdot 1}$

Step 3.5.2

Multiply $2$ by $1$ .

$r = \frac{- 4 \pm 2 \sqrt{2 (x + 3)}}{2}$

Step 3.5.3

Simplify $\frac{- 4 \pm 2 \sqrt{2 (x + 3)}}{2}$ .

$r = - 2 \pm \sqrt{2 (x + 3)}$

Step 3.5.4

Change the $\pm$ to $+$ .

$r = - 2 + \sqrt{2 (x + 3)}$

Step 3.6

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

Raise $4$ to the power of $2$ .

$r = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot (- 2 x - 2)}}{2 \cdot 1}$

Step 3.6.1.2

Multiply $- 4$ by $1$ .

$r = \frac{- 4 \pm \sqrt{16 - 4 \cdot (- 2 x - 2)}}{2 \cdot 1}$

Step 3.6.1.3

Apply the distributive property.

$r = \frac{- 4 \pm \sqrt{16 - 4 (- 2 x) - 4 \cdot - 2}}{2 \cdot 1}$

Step 3.6.1.4

Multiply $- 2$ by $- 4$ .

$r = \frac{- 4 \pm \sqrt{16 + 8 x - 4 \cdot - 2}}{2 \cdot 1}$

Step 3.6.1.5

Multiply $- 4$ by $- 2$ .

$r = \frac{- 4 \pm \sqrt{16 + 8 x + 8}}{2 \cdot 1}$

Step 3.6.1.6

Add $16$ and $8$ .

$r = \frac{- 4 \pm \sqrt{8 x + 24}}{2 \cdot 1}$

Step 3.6.1.7

Factor $8$ out of $8 x + 24$ .

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Step 3.6.1.7.1

Factor $8$ out of $8 x$ .

$r = \frac{- 4 \pm \sqrt{8 (x) + 24}}{2 \cdot 1}$

Step 3.6.1.7.2

Factor $8$ out of $24$ .

$r = \frac{- 4 \pm \sqrt{8 x + 8 \cdot 3}}{2 \cdot 1}$

Step 3.6.1.7.3

Factor $8$ out of $8 x + 8 \cdot 3$ .

$r = \frac{- 4 \pm \sqrt{8 (x + 3)}}{2 \cdot 1}$

Step 3.6.1.8

Rewrite $8 (x + 3)$ as $2^{2} \cdot (2 (x + 3))$ .

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Step 3.6.1.8.1

Factor $4$ out of $8$ .

$r = \frac{- 4 \pm \sqrt{4 (2) (x + 3)}}{2 \cdot 1}$

Step 3.6.1.8.2

Rewrite $4$ as $2^{2}$ .

$r = \frac{- 4 \pm \sqrt{2^{2} \cdot (2 (x + 3))}}{2 \cdot 1}$

Step 3.6.1.8.3

Add parentheses.

$r = \frac{- 4 \pm \sqrt{2^{2} \cdot (2 (x + 3))}}{2 \cdot 1}$

Step 3.6.1.9

Pull terms out from under the radical.

$r = \frac{- 4 \pm 2 \sqrt{2 (x + 3)}}{2 \cdot 1}$

Step 3.6.2

Multiply $2$ by $1$ .

$r = \frac{- 4 \pm 2 \sqrt{2 (x + 3)}}{2}$

Step 3.6.3

Simplify $\frac{- 4 \pm 2 \sqrt{2 (x + 3)}}{2}$ .

$r = - 2 \pm \sqrt{2 (x + 3)}$

Step 3.6.4

Change the $\pm$ to $-$ .

$r = - 2 - \sqrt{2 (x + 3)}$

Step 3.7

The final answer is the combination of both solutions.

$r = - 2 + \sqrt{2 (x + 3)}$

$r = - 2 - \sqrt{2 (x + 3)}$

$r = - 2 + \sqrt{2 (x + 3)}$

$r = - 2 - \sqrt{2 (x + 3)}$

Step 4

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

$y_{1} = e^{(- 2 + \sqrt{2 (x + 3)}) x}$

$y_{2} = e^{(- 2 - \sqrt{2 (x + 3)}) x}$

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.

$y = c_{1} e^{(- 2 + \sqrt{2 (x + 3)}) x} + c_{2} e^{(- 2 - \sqrt{2 (x + 3)}) x}$

Step 6

Simplify each term.

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Step 6.1

Apply the distributive property.

$y = c_{1} e^{- 2 x + \sqrt{2 (x + 3)} x} + c_{2} e^{(- 2 - \sqrt{2 (x + 3)}) x}$

Step 6.2

Apply the distributive property.

$y = c_{1} e^{- 2 x + \sqrt{2 (x + 3)} x} + c_{2} e^{- 2 x - \sqrt{2 (x + 3)} x}$