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Question Number 117608 by TANMAY PANACEA last updated on 12/Oct/20

solve  (d^2 y/dt^2 )+w^2 x=0

$${solve} \\ $$$$\frac{{d}^{\mathrm{2}} {y}}{{dt}^{\mathrm{2}} }+{w}^{\mathrm{2}} {x}=\mathrm{0} \\ $$

Answered by Dwaipayan Shikari last updated on 12/Oct/20

(d/dt)((dy/dt))=−w^2 x  (da/dt)=−w^2 x        a=(dy/dt)  a=−w^2 xt+C  (dy/dt)=−w^2 xt+C⇒y=−(w^2 /2)xt^2 +Ct+A

$$\frac{{d}}{{dt}}\left(\frac{{dy}}{{dt}}\right)=−{w}^{\mathrm{2}} {x} \\ $$$$\frac{{da}}{{dt}}=−{w}^{\mathrm{2}} {x}\:\:\:\:\:\:\:\:{a}=\frac{{dy}}{{dt}} \\ $$$${a}=−{w}^{\mathrm{2}} {xt}+{C} \\ $$$$\frac{{dy}}{{dt}}=−{w}^{\mathrm{2}} {xt}+{C}\Rightarrow{y}=−\frac{{w}^{\mathrm{2}} }{\mathrm{2}}{xt}^{\mathrm{2}} +{Ct}+{A} \\ $$

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