# Form the differential equuations by eliminating arbitary constants given in brackets against each. $y= e^{3x}(C \cos 2x +D \sin 2x ) [C , D ]$

Toolbox:
• If we have an equation $f(x,y,c_1,c_2,....c_n)=u$ Containing n arbitrary constant $c_1,c_2...c_n$, then by differentiating n times, we get $(n+1)$ equations in total. If we eliminate the arbitrary constants $c_1,c_2....c_n,$ we get a D.E of order n
Given $y= e^{3x}[C \cos 2x+D \sin 2x]$-----(1)
Step 1:
$\large\frac{dy}{dx}$$=3e^{3x}[C \cos 2x+D \sin 2x]+e^{3x}[-2C \sin 2+ 2D \cos 2x] \therefore \large\frac{dy}{dx}$$=3y+2e^{3x}[-C \sin 2x+ D \cos 2x]$-----(ii)
Step 2:
$\large\frac{d^2y}{dx^2}$$=3 \large\frac{dy}{dx} \normalsize +6e^{3x}[-C \sin 2x+D \cos 2x]+4e^{3x}[-2C \cos 2x- 2D \sin 2x] Step 3: \large\frac{d^2y}{dx^2}$$=3\large \frac{dy}{dx}$$+6e^{3x}[-C \sin 2x+D \cos 2x]-4y-----(iii) Step 4: From (ii) 2e^{3x}(-c \sin 2x +D \cos 2x)= \large\frac{dy}{dx}$$-3y$
Step 5:
Substitute in (iii)
$\large\frac{d^2y}{dx^2}$$=3\large \frac{dy}{dx}$$+3\bigg[\large\frac{dy}{dx}$$-3y\bigg]-4y \large\frac{d^2y}{dx^2}$$-6\large \frac{dy}{dx}$$+13y=0$ is the required D.E