If y = Ae*** cos (pt + c), prove th...
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If y = Ae*** cos (pt + c), prove that + 2 y +n’y=0, where n?= p +K?. dt

11th - 12th Class
Maths
Solution
122
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( y=A e^{-k t} cos (p t+c) ) Differentiating w.r.t. ( t, ) we get ( frac{d y}{d t}=-k A e^{-k t} cos (p t+c)-p A e^{-k t} sin (p t+c) ) ( Rightarrow frac{d y}{d t}=-k y-p A e^{-k t} sin (p t+c) ) Differentiating again w.r.t. ( t, ) we get ( frac{d^{p} y}{d t^{p}}=-k frac{d y}{d x}+p k A e^{-k t} sin (p t+c)-p^{2} A e^{-k t} cos (p t+c) ) ( Rightarrow frac{d^{p} y}{d t^{p}}=-k frac{d y}{d x}+p k A e^{-k t} sin (p t+c)-p^{2} y ) ( Rightarrow frac{d^{p} y}{d t^{2}}=-k frac{d y}{d x}+kleft(-k y-frac{d y}{d x}right)-p^{2} y quad ) From ( Rightarrow frac{d^{2} y}{d t^{p}}=-k frac{d y}{d x}-k^{2} y-k frac{d y}{d x}-p^{2} y ) ( Rightarrow frac{d^{p} y}{d t^{p}}=-2 k frac{d y}{d x}-left(k^{2}+p^{2}right) y ) ( Rightarrow frac{d^{p} y}{d t^{p}}+2 k frac{d y}{d x}+left(k^{2}+p^{2}right) y=0 ) ( Rightarrow frac{d^{p} boldsymbol{y}}{d t^{p}}+2 k frac{d y}{d x}+n^{2} y=0 quad ) where ( n^{2}=k^{2}+p^{2} )
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