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Form the differential equuations by eliminating arbitary constants given in brackets against each. $y=Ae^{2x} \cos (3x , +B ) [A , B ]$

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  • 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
Step 1:
$y= Ae^{2x} \cos (3x+B)$-----(i)$
$\large\frac{dy}{dx}$$=2 Ae^{2x} \cos (3x+B)-Ae^{2x}. 3 \sin (3x+B)$
(ie) $\large\frac{dy}{dx}$$=2y-3Ae^{2x} \sin (3x+B) $ -----(ii)
Step 2:
From (ii) $3Ae^{2x} \sin (3x+B)=2y- \large\frac{dy}{dx}$-----(iii)
Step 3:
Differentiate (ii) again
(ie) $\large\frac{d^2y}{dx^2}$$=2\large\frac{dy}{dx}$$-6Ae^{2x} \sin (3x+B)-3 Ae^{2x}, 3 \cos (3x+3)$-(ii)
(ie) $\large\frac{d^2y}{dx^2}$$=2\large\frac{dy}{dx}$$-2(2y -\large\frac{dy}{dx})$$-9y$ from (iii)
$\large\frac{d^2y}{dx^2}$$-4\large\frac{dy}{dx}$$+13y=0$ is the required D.E
answered Sep 5, 2013 by meena.p

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