Graph the particular solution and several other solutions. $2 y ^ { \prime \prime } + 3 y ^ { \prime } + y = 2 + \cos 2 x$

Question

Quizplus · Accepted Answer

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

$$ 2 y ^ { \prime \prime } + 3 y ^ { \prime } + y = 0 $$

The characteristic equation is

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

which has roots $r=-\frac{1}{2},-1$. Therefore, the general solution to the homogeneous equation is

$$ y_h = c_1 e^{-\frac{1}{2}x} + c_2 e^{-x} $$

Next, we find a particular solution to the nonhomogeneous equation using the method of undetermined coefficients. Since the right-hand side is a sum of a constant and a cosine function, we guess a particular solution of the form

$$ y_p = A + B\cos(2x) $$

where $A$ and $B$ are constants to be determined. Then, we have

\begin{align*}
y_p' &= 0 - 2B\sin(2x) \
y_p'' &= 0 - 4B\cos(2x)
\end{align*}

Substituting into the differential equation, we get

$$ -4B\cos(2x) + 0 - B\cos(2x) = 2 + \cos(2x) $$

Solving for $B$, we get $B=-\frac{1}{5}$. Substituting back, we can find $A=\frac{77}{50}$. Therefore, the particular solution is

$$ y_p = \frac{77}{50} - \frac{1}{5}\cos(2x) $$

The general solution to the nonhomogeneous equation is the sum of the homogeneous solution and the particular solution:

$$ y = y_h + y_p = c_1 e^{-\frac{1}{2}x} + c_2 e^{-x} + \frac{77}{50} - \frac{1}{5}\cos(2x) $$

To graph the particular solution and several other solutions, we can choose any values for $c_1$ and $c_2$. In the graph below, we choose $c_1=2$ and $c_2=-1$ to obtain three different solutions:

\includegraphics[width=8cm]{plot.png}

Graph the Particular Solution and Several Other Solutions 2y′′+3y′+y=2+cos⁡2x2 y ^ { \prime \prime } + 3 y ^ { \prime } + y = 2 + \cos 2 x2y′′+3y′+y=2+cos2x

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