Graph the particular solution and several other solutions. Select the correct answer.
$$2 y ^ { tt } + 3 y ^ { t } + y = 2 + \cos 2 x$$

Question

Quizplus · Accepted Answer

To graph the particular solution and several other solutions, we need to first solve the homogeneous equation:

$$2 y ^ { tt } + 3 y ^ { t } + y = 0$$

The characteristic equation is:

$$2 r ^ 2 + 3 r + 1 = 0$$

which factors as

$$(2r + 1)(r+1) = 0$$

so the roots are $r = - \frac{1}{2}$ and $r = -1$. Therefore, the general solution to the homogeneous equation is:

$$y_h = c_1 e^{-x/2} + c_2 e^{-x}$$

To find a particular solution to the non-homogeneous equation, we assume that the solution has the form:

$$y_p = A + B \cos 2x + C \sin 2x$$

Taking first and second derivatives, we get:

$$ y_p^{\prime} = 2B \sin 2x - 2C \cos 2x$$
$$ y_p^{\prime \prime} = -4B \cos 2x - 4C \sin 2x$$

Substituting these into the original equation gives:

$$-4B + 3(2B \sin 2x - 2C \cos 2x) + (A + B \cos 2x + C \sin 2x) = 2 + \cos 2x$$

Simplifying this expression, we get:

$$(3B + C) \sin 2x - (3A - B) \cos 2x = 6B - 1$$

Since $\sin 2x$ and $\cos 2x$ are linearly independent, the coefficients must be equal on both sides. Therefore, we have the system of equations:

$$3B + C = 0$$ 
$$3A - B = -1$$ 
$$6B - 1 = 2$$

Solving for $A$, $B$, and $C$, we get:

$$A = -\frac{1}{3}, \quad B = \frac{1}{6}, \quad C = -\frac{1}{2}$$

So the particular solution is:

$$y_p = -\frac{1}{3} + \frac{1}{6} \cos 2x - \frac{1}{2} \sin 2x$$

Therefore, the general solution to the non-homogeneous equation is:

$$y = y_h + y_p = c_1 e^{-x/2} + c_2 e^{-x} - \frac{1}{3} + \frac{1}{6} \cos 2x - \frac{1}{2} \sin 2x$$

We can graph this function using a graphing calculator or software. To get several other solutions, we can vary the constants $c_1$ and $c_2$.

Option B shows a graph that matches the solution we just found. Option A is the graph of the homogeneous solution, and option C is not a graph of a solution to the equation.

Graph the Particular Solution and Several Other Solutions 2ytt+3yt+y=2+cos⁡2x2 y ^ { tt } + 3 y ^ { t } + y = 2 + \cos 2 x2ytt+3yt+y=2+cos2x

Graph the Particular Solution and Several Other Solutions $2 y ^ { tt } + 3 y ^ { t } + y = 2 + \cos 2 x$