Calculus Examples

$\frac{d^{2} y}{d x^{2}} + 4 \frac{d y}{d x} - 2 y = 2 x^{2} - 3 x + 6$

Step 1

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

Step 2

Find the characteristic equation for $\frac{d^{2} y}{d x^{2}} + 4 \frac{d y}{d x} - 2 y = 2 x^{2} - 3 x + 6$ .

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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.

$r^{2} e^{r x} + 4 (r e^{r x}) - 2 e^{r x} = 2 x^{2} - 3 x + 6$

Step 2.4

Remove parentheses.

$r^{2} e^{r x} + 4 r e^{r x} - 2 e^{r x} = 2 x^{2} - 3 x + 6$

Step 2.5

Factor out $e^{r x}$ .

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Step 2.5.1

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

$e^{r x} r^{2} + 4 r e^{r x} - 2 e^{r x} = 2 x^{2} - 3 x + 6$

Step 2.5.2

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

$e^{r x} r^{2} + e^{r x} (4 r) - 2 e^{r x} = 2 x^{2} - 3 x + 6$

Step 2.5.3

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

$e^{r x} (r^{2} + 4 r) - 2 e^{r x} = 2 x^{2} - 3 x + 6$

Step 2.5.4

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

$e^{r x} (r^{2} + 4 r) + e^{r x} \cdot - 2 = 2 x^{2} - 3 x + 6$

Step 2.5.5

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

$e^{r x} (r^{2} + 4 r - 2) = 2 x^{2} - 3 x + 6$

Step 2.6

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

$r^{2} + 4 r - 2 = 2 x^{2} - 3 x + 6$

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 $2 x^{2}$ from both sides of the equation.

$r^{2} + 4 r - 2 - 2 x^{2} = - 3 x + 6$

Step 3.1.1.2

Add $3 x$ to both sides of the equation.

$r^{2} + 4 r - 2 - 2 x^{2} + 3 x = 6$

Step 3.1.1.3

Subtract $6$ from both sides of the equation.

$r^{2} + 4 r - 2 - 2 x^{2} + 3 x - 6 = 0$

Step 3.1.2

Subtract $6$ from $- 2$ .

$r^{2} + 4 r - 2 x^{2} + 3 x - 8 = 0$

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 $c = - 2 x^{2} + 3 x - 8$ into the quadratic formula and solve for $r$ .

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

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 $2$ .

$r = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot (- 2 x^{2} + 3 x - 8)}}{2 \cdot 1}$

Step 3.4.1.2

Multiply $- 4$ by $1$ .

$r = \frac{- 4 \pm \sqrt{16 - 4 \cdot (- 2 x^{2} + 3 x - 8)}}{2 \cdot 1}$

Step 3.4.1.3

Apply the distributive property.

$r = \frac{- 4 \pm \sqrt{16 - 4 (- 2 x^{2}) - 4 (3 x) - 4 \cdot - 8}}{2 \cdot 1}$

Step 3.4.1.4

Simplify.

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Step 3.4.1.4.1

Multiply $- 2$ by $- 4$ .

$r = \frac{- 4 \pm \sqrt{16 + 8 x^{2} - 4 (3 x) - 4 \cdot - 8}}{2 \cdot 1}$

Step 3.4.1.4.2

Multiply $3$ by $- 4$ .

$r = \frac{- 4 \pm \sqrt{16 + 8 x^{2} - 12 x - 4 \cdot - 8}}{2 \cdot 1}$

Step 3.4.1.4.3

Multiply $- 4$ by $- 8$ .

$r = \frac{- 4 \pm \sqrt{16 + 8 x^{2} - 12 x + 32}}{2 \cdot 1}$

Step 3.4.1.5

Add $16$ and $32$ .

$r = \frac{- 4 \pm \sqrt{8 x^{2} - 12 x + 48}}{2 \cdot 1}$

Step 3.4.1.6

Factor $4$ out of $8 x^{2} - 12 x + 48$ .

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Step 3.4.1.6.1

Factor $4$ out of $8 x^{2}$ .

$r = \frac{- 4 \pm \sqrt{4 (2 x^{2}) - 12 x + 48}}{2 \cdot 1}$

Step 3.4.1.6.2

Factor $4$ out of $- 12 x$ .

$r = \frac{- 4 \pm \sqrt{4 (2 x^{2}) + 4 (- 3 x) + 48}}{2 \cdot 1}$

Step 3.4.1.6.3

Factor $4$ out of $48$ .

$r = \frac{- 4 \pm \sqrt{4 (2 x^{2}) + 4 (- 3 x) + 4 (12)}}{2 \cdot 1}$

Step 3.4.1.6.4

Factor $4$ out of $4 (2 x^{2}) + 4 (- 3 x)$ .

$r = \frac{- 4 \pm \sqrt{4 (2 x^{2} - 3 x) + 4 (12)}}{2 \cdot 1}$

Step 3.4.1.6.5

Factor $4$ out of $4 (2 x^{2} - 3 x) + 4 (12)$ .

$r = \frac{- 4 \pm \sqrt{4 (2 x^{2} - 3 x + 12)}}{2 \cdot 1}$

Step 3.4.1.7

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

$r = \frac{- 4 \pm \sqrt{2^{2} (2 x^{2} - 3 x + 12)}}{2 \cdot 1}$

Step 3.4.1.8

Pull terms out from under the radical.

$r = \frac{- 4 \pm 2 \sqrt{2 x^{2} - 3 x + 12}}{2 \cdot 1}$

Step 3.4.2

Multiply $2$ by $1$ .

$r = \frac{- 4 \pm 2 \sqrt{2 x^{2} - 3 x + 12}}{2}$

Step 3.4.3

Simplify $\frac{- 4 \pm 2 \sqrt{2 x^{2} - 3 x + 12}}{2}$ .

$r = - 2 \pm \sqrt{2 x^{2} - 3 x + 12}$

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 $2$ .

$r = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot (- 2 x^{2} + 3 x - 8)}}{2 \cdot 1}$

Step 3.5.1.2

Multiply $- 4$ by $1$ .

$r = \frac{- 4 \pm \sqrt{16 - 4 \cdot (- 2 x^{2} + 3 x - 8)}}{2 \cdot 1}$

Step 3.5.1.3

Apply the distributive property.

$r = \frac{- 4 \pm \sqrt{16 - 4 (- 2 x^{2}) - 4 (3 x) - 4 \cdot - 8}}{2 \cdot 1}$

Step 3.5.1.4

Simplify.

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Step 3.5.1.4.1

Multiply $- 2$ by $- 4$ .

$r = \frac{- 4 \pm \sqrt{16 + 8 x^{2} - 4 (3 x) - 4 \cdot - 8}}{2 \cdot 1}$

Step 3.5.1.4.2

Multiply $3$ by $- 4$ .

$r = \frac{- 4 \pm \sqrt{16 + 8 x^{2} - 12 x - 4 \cdot - 8}}{2 \cdot 1}$

Step 3.5.1.4.3

Multiply $- 4$ by $- 8$ .

$r = \frac{- 4 \pm \sqrt{16 + 8 x^{2} - 12 x + 32}}{2 \cdot 1}$

Step 3.5.1.5

Add $16$ and $32$ .

$r = \frac{- 4 \pm \sqrt{8 x^{2} - 12 x + 48}}{2 \cdot 1}$

Step 3.5.1.6

Factor $4$ out of $8 x^{2} - 12 x + 48$ .

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Step 3.5.1.6.1

Factor $4$ out of $8 x^{2}$ .

$r = \frac{- 4 \pm \sqrt{4 (2 x^{2}) - 12 x + 48}}{2 \cdot 1}$

Step 3.5.1.6.2

Factor $4$ out of $- 12 x$ .

$r = \frac{- 4 \pm \sqrt{4 (2 x^{2}) + 4 (- 3 x) + 48}}{2 \cdot 1}$

Step 3.5.1.6.3

Factor $4$ out of $48$ .

$r = \frac{- 4 \pm \sqrt{4 (2 x^{2}) + 4 (- 3 x) + 4 (12)}}{2 \cdot 1}$

Step 3.5.1.6.4

Factor $4$ out of $4 (2 x^{2}) + 4 (- 3 x)$ .

$r = \frac{- 4 \pm \sqrt{4 (2 x^{2} - 3 x) + 4 (12)}}{2 \cdot 1}$

Step 3.5.1.6.5

Factor $4$ out of $4 (2 x^{2} - 3 x) + 4 (12)$ .

$r = \frac{- 4 \pm \sqrt{4 (2 x^{2} - 3 x + 12)}}{2 \cdot 1}$

Step 3.5.1.7

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

$r = \frac{- 4 \pm \sqrt{2^{2} (2 x^{2} - 3 x + 12)}}{2 \cdot 1}$

Step 3.5.1.8

Pull terms out from under the radical.

$r = \frac{- 4 \pm 2 \sqrt{2 x^{2} - 3 x + 12}}{2 \cdot 1}$

Step 3.5.2

Multiply $2$ by $1$ .

$r = \frac{- 4 \pm 2 \sqrt{2 x^{2} - 3 x + 12}}{2}$

Step 3.5.3

Simplify $\frac{- 4 \pm 2 \sqrt{2 x^{2} - 3 x + 12}}{2}$ .

$r = - 2 \pm \sqrt{2 x^{2} - 3 x + 12}$

Step 3.5.4

Change the $\pm$ to $+$ .

$r = - 2 + \sqrt{2 x^{2} - 3 x + 12}$

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 $2$ .

$r = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot (- 2 x^{2} + 3 x - 8)}}{2 \cdot 1}$

Step 3.6.1.2

Multiply $- 4$ by $1$ .

$r = \frac{- 4 \pm \sqrt{16 - 4 \cdot (- 2 x^{2} + 3 x - 8)}}{2 \cdot 1}$

Step 3.6.1.3

Apply the distributive property.

$r = \frac{- 4 \pm \sqrt{16 - 4 (- 2 x^{2}) - 4 (3 x) - 4 \cdot - 8}}{2 \cdot 1}$

Step 3.6.1.4

Simplify.

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Step 3.6.1.4.1

Multiply $- 2$ by $- 4$ .

$r = \frac{- 4 \pm \sqrt{16 + 8 x^{2} - 4 (3 x) - 4 \cdot - 8}}{2 \cdot 1}$

Step 3.6.1.4.2

Multiply $3$ by $- 4$ .

$r = \frac{- 4 \pm \sqrt{16 + 8 x^{2} - 12 x - 4 \cdot - 8}}{2 \cdot 1}$

Step 3.6.1.4.3

Multiply $- 4$ by $- 8$ .

$r = \frac{- 4 \pm \sqrt{16 + 8 x^{2} - 12 x + 32}}{2 \cdot 1}$

Step 3.6.1.5

Add $16$ and $32$ .

$r = \frac{- 4 \pm \sqrt{8 x^{2} - 12 x + 48}}{2 \cdot 1}$

Step 3.6.1.6

Factor $4$ out of $8 x^{2} - 12 x + 48$ .

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Step 3.6.1.6.1

Factor $4$ out of $8 x^{2}$ .

$r = \frac{- 4 \pm \sqrt{4 (2 x^{2}) - 12 x + 48}}{2 \cdot 1}$

Step 3.6.1.6.2

Factor $4$ out of $- 12 x$ .

$r = \frac{- 4 \pm \sqrt{4 (2 x^{2}) + 4 (- 3 x) + 48}}{2 \cdot 1}$

Step 3.6.1.6.3

Factor $4$ out of $48$ .

$r = \frac{- 4 \pm \sqrt{4 (2 x^{2}) + 4 (- 3 x) + 4 (12)}}{2 \cdot 1}$

Step 3.6.1.6.4

Factor $4$ out of $4 (2 x^{2}) + 4 (- 3 x)$ .

$r = \frac{- 4 \pm \sqrt{4 (2 x^{2} - 3 x) + 4 (12)}}{2 \cdot 1}$

Step 3.6.1.6.5

Factor $4$ out of $4 (2 x^{2} - 3 x) + 4 (12)$ .

$r = \frac{- 4 \pm \sqrt{4 (2 x^{2} - 3 x + 12)}}{2 \cdot 1}$

Step 3.6.1.7

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

$r = \frac{- 4 \pm \sqrt{2^{2} (2 x^{2} - 3 x + 12)}}{2 \cdot 1}$

Step 3.6.1.8

Pull terms out from under the radical.

$r = \frac{- 4 \pm 2 \sqrt{2 x^{2} - 3 x + 12}}{2 \cdot 1}$

Step 3.6.2

Multiply $2$ by $1$ .

$r = \frac{- 4 \pm 2 \sqrt{2 x^{2} - 3 x + 12}}{2}$

Step 3.6.3

Simplify $\frac{- 4 \pm 2 \sqrt{2 x^{2} - 3 x + 12}}{2}$ .

$r = - 2 \pm \sqrt{2 x^{2} - 3 x + 12}$

Step 3.6.4

Change the $\pm$ to $-$ .

$r = - 2 - \sqrt{2 x^{2} - 3 x + 12}$

Step 3.7

The final answer is the combination of both solutions.

$r = - 2 + \sqrt{2 x^{2} - 3 x + 12}$

$r = - 2 - \sqrt{2 x^{2} - 3 x + 12}$

$r = - 2 + \sqrt{2 x^{2} - 3 x + 12}$

$r = - 2 - \sqrt{2 x^{2} - 3 x + 12}$

Step 4

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

$y_{1} = e^{(- 2 + \sqrt{2 x^{2} - 3 x + 12}) x}$

$y_{2} = e^{(- 2 - \sqrt{2 x^{2} - 3 x + 12}) x}$

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.

$y = c_{1} e^{(- 2 + \sqrt{2 x^{2} - 3 x + 12}) x} + c_{2} e^{(- 2 - \sqrt{2 x^{2} - 3 x + 12}) x}$

Step 6

Simplify each term.

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Step 6.1

Apply the distributive property.

$y = c_{1} e^{- 2 x + \sqrt{2 x^{2} - 3 x + 12} x} + c_{2} e^{(- 2 - \sqrt{2 x^{2} - 3 x + 12}) x}$

Step 6.2

Apply the distributive property.

$y = c_{1} e^{- 2 x + \sqrt{2 x^{2} - 3 x + 12} x} + c_{2} e^{- 2 x - \sqrt{2 x^{2} - 3 x + 12} x}$