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Form the differential equuations by eliminating arbitary constants given in brackets against each. $ y=Ae^{2x}+Be^{-5x} [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
$y=Ae^{2x}+Be^{-5x}$----(i)
Step 1:
$\large\frac{dy}{dx}$$=2Ae^{2x}-5Be^{-5x}$
$\qquad= 2Ae^{2x}+2Be^{-5x}-7Be^{-5x}$
$\qquad=2y-7Be^{-5x}$-----(ii)
From (ii) $7Be^{-5x}=2y-\large\frac{dy}{dx}$ -----(iii)
Step 2:
Differentiating (ii) again w.r.t x
$\large\frac{d^2y}{dx^2}$$=2 \large\frac{dy}{dx}$$+35 Be^{-5x}$-----(iv)
Step 3:
Substituting in (iv) from (iii)
$\large\frac{d^2y}{dx^2}$$=2 \large \frac{dy}{dx}$$+5 [2y- \large\frac{dy}{dx}]$
$\large\frac{d^2y}{dx^2}$$+3\large\frac{dy}{dx}$$-10y=0$ is the DE
answered Sep 5, 2013 by meena.p
 

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