To check whether y=c1ex+c2e2x is the solution of y′′−2y+3y=0...(i)
Now, y=c1ex+c2e2x
y′=c1ex+2c2e2xy′′=c1ex+4c2e2x
Put these in LHS of (i),
=y′′−2y+3y=c1ex+4c2e2x−2(c1ex+2c2e2x)+3(c1ex+c2e2x)=2c1ex+3c2e2x=0
So, it is not the solution of given D.E.
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