If $$F_{\mathrm{ext}}(t)=0$$, equation (3) becomes

$$m y^{\prime \prime}+b y^{\prime}+k y=0$$

For this equation, verify the following:

(a) If $$y(t)$$ is a solution, so is $$c y(t)$$, for any constant $$C$$.
(b) If $$y_{1}(t)$$ and $$y_{2}(t)$$ are solutions,so is their sum $$y_{1}(t)+y_{2}(t)$$.

Answer

a. $c\left(m y^{\prime \prime}+b y^{\prime}+k y\right)=0$
b. $m\left[y_{1}(t)+y_{2}(t)\right]^{\prime \prime}+b\left[y_{1}(t)+y_{2}(t)\right]^{\prime}+k\left[y_{1}(t)+y_{2}(t)\right]=0=m y^{\prime \prime}+b y^{\prime}+k y=0$

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Video Transcript

okay, Were Astra showing this problem if m y prime Prime plus B y prime plus K Y is equal zero is the equation given to us with the external force of the right hand side equal to zero so f external equal to zero, then we want to show that if why solve this equation, then sort of see why and sodas, Why one plus y two. Where Why wouldn't want to her functions of Pete and why is equal to y one plus Wait, too. Let's make this white Tilda and this is why. Yeah, So you want to show that Why hat and White? Till they're also solutions of this equation. Now notice that M times why had Prime Prime plus b times y hat time. Let's k Time's white hat is equal to m times. See why crime print plus b times see why prime plus k times see why and the sea factor sort of the whole equation. And since this is equal to zero has given above, does it turn things equal to zero? And now, for why? Tilda we have em. Why Tilda Prime Crime Plus B why I told her prime plus k Y Tilda is equal to m times. Why one prime Prime plus y two prime prime was be times Why one prime, That's why to prime plus k times. Why one plus white too. And since why one and why tour Given as solutions to the differential equation we have that we could break this up across the plus sides and we have em. Why one prime prime? That's not a dog plus M. Why to prime time start or don't is equal to is equal to 00 which is equal to zero. So we have shown what we wanted to show.

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