Question

$ y=2{{e}^{2x}}-{{e}^{-x}} $ is a solution of the differential equation

• A. $ {{y}_{2}}+{{y}_{1}}+2y=0 $
• B. $ {{y}_{2}}-{{y}_{1}}+2y=0 $
• C. $ {{y}_{2}}+{{y}_{1}}=0 $
• D. $ {{y}_{2}}-{{y}_{1}}-2y=0 $
• E. $ {{y}_{2}}-2{{y}_{1}}-y=0 $

Answer 1

Answer: (D)

Given that, $ y=2{{e}^{2x}}-{{e}^{-x}} $
On differentiating w.r.t. $ x, $ we get $ {{y}_{1}}=4{{e}^{2x}}+{{e}^{-x}} $
Now, on again differentiating w.r.t. $ x, $
we get $ {{y}_{2}}=8{{e}^{2x}}-{{e}^{-x}} $ Now, $ {{y}_{2}}-{{y}_{1}}-2y $
$=8{{e}^{2x}}-{{e}^{-x}}-4{{e}^{2x}}-{{e}^{-x}}-4{{e}^{2x}}+2{{e}^{-x}} $
$=0 $