Finding particular integral to $ (D^3 + 4D)y = \sin2x$ The question is to find particular integral of this differential equation 
$$ (D^3 + 4D)y = \sin2x.$$
where $$ D = \frac{d}{dx} $$
Please looking for your quick tip guys.
 A: the first step is finding the complementary solution 
$$D(D^2+4)=0$$
$$D=0$$
$$D=\pm2i$$
so the complementary solution is
$$y_c=c_1+c_2\cos2x+c_3\sin 2x$$
now assume the particular solution as follow
$$y_p=A\cos 2x+B\sin 2x$$
because the similarity with complementary solution, we should multiply the particular solution by $x$, so
$$y_p=x(A\cos 2x+B\sin 2x)$$
then you can complete the solution     
A: We need to find a particular solution to the following ODE:
$$(D^3 + 4D)y = \sin2x$$
Or, in the more clear notation:
$$y'''+4y' = \sin2x$$
On the first step we introduce a new function $u=y'$.

$$u''+4u = \sin2x$$

Now, this is forced harmonic oscillator. There are a lot of ways to obtain the general (or particular) solution, but I will use a general brute force approach.
Let's represent $u$ in the form of the solution to the homogenous equation, but let the constants depend on $x$:
$$u=A(x) \cos (2x)+B(x) \sin (2x)$$
Now we can explicitly compute the second derivative to find:

$$u''+4u=(A''+4B') \cos (2x)+(B''-4A') \sin (2x)$$

Comparing this to the original equation we have:
$$\begin{cases} A''+4B'=0 \\ B''-4A'=1 \end{cases}$$
Since we only need a particular solution, we can guess a simple one:
$$B=C_1=\text{const}, \qquad A=C_2-\frac{x}{4}=\text{const}-\frac{x}{4}$$
Setting $C_1=C_2=0$ we get a particular solution:

$$u_p=-\frac{x}{4} \cos (2x)$$

Integrating one time, we get a final answer:

$$y_p=-\frac{1}{16} (2x \sin (2x)+\cos(2x))$$

A: Given:- (D³+4D)y=sin2x
Condition : 1/f(D), if after equal to sinax or cosax given means you have to replace D² into -(a²)
Ans-  (1/D³+4D)*sin2x
[1/D(D²+4)]*sin2x    Here, D² is replaced by -(a²) that is
D²= -(2²)
[1/D(-4+4)]*sin2x  (important first square the number and put (-) symbol.)
[1/D(0)]*sin2x [1/0]*sinx ( condition if denomerator  is zero multiply  numerator by x and differentiate f(D)).
[x/3D²+4]*sin2x ( substitute D² value)
[x/3(-4)+4]*sin2x [x/(-12+4)]*sin2x
Therefore [-x/8]*sin2x.
