An external force $$F(t)=2 \cos 2 t$$ is applied to a mass spring system with $$m=1, b=0$$ and $$k=4$$, which is initially at rest; i.e., $$y(0)=0, y^{\prime}(0)=0$$. Verify that $$y(t)=\frac{1}{2} t \sin 2 t$$ gives the motion of this spring.What will eventually (as t increases) happen to the spring?

Answer

$\lim _{t \rightarrow \infty} \frac{y}{x}(t)=\infty$

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Calculus 2 / BC

Fundamentals of Differential Equations

Chapter 4

Linear Second-Order Equations

Section 1

Introduction: The Mass-Spring Oscillator

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Video Transcript

Okay, so we have our external force here for the differential equation is to co sign to t and we have the initial conditions at rest given by, uh, why was your equal zero and why prime of zero equals zero. So these are the initial conditions for the spring and the differential equation Describing the spring using the values for M B and key is why Prime Prime Plus Four Why equals to co sign to t and we have The particular solution is given us why P equals 1/2 t sign to t. So when we differentiate this twice and then added to four times itself, we will we can verify and see that why people in front was four y p is equal to co sign to t. And you can also check with this Y p formulation that the initial conditions are also satisfied. And we also have that the limit s t goes to infinity off why of tea or why P of tea in this case is equal to infinity because of the linear dependence on tea in the solution. So the spring will break down in the limit of large oscillations

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