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The DE obtained by eliminating $c_1$ and $c_2$ from $y = c_1 e^{3x} + c_2 e^{2x}$ is:
A$d^2y/dx^2 - 5\,dy/dx + 6y = 0$
B$d^2y/dx^2 + 5\,dy/dx + 6y = 0$
C$d^2y/dx^2 - dy/dx + 6y = 0$
D$d^2y/dx^2 + 6y = 0$
Answer & Solution
Correct answer: A. $d^2y/dx^2 - 5\,dy/dx + 6y = 0$
Characteristic roots 3 and 2 give the auxiliary equation $(D-3)(D-2) = 0$ ⇒ $D^2 - 5D + 6 = 0$, i.e., $d^2y/dx^2 - 5\,dy/dx + 6y = 0$. (Alternative: differentiate twice and eliminate $c_1, c_2$.)
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