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Find the motion of the mass-spring system modeled by the ODE and initial conditions. Sketch or graph the solution curve. In addition, sketch or graph the curve of yypy-y_p to see when the system practically reaches the steady state.

(D2+2D+I)y=75(sint12sin2t+13sin3t)\left(D^2+2 D+I\right) y=75\left(\sin t-\frac{1}{2} \sin 2 t+\frac{1}{3} \sin 3 t\right), y(0)=0.y(0)=1y(0)=0 . \quad y^{\prime}(0)=1


Answered 2 years ago
Answered 2 years ago
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We are actually given the following equation:

y+2y+y=75(sint12sin2t+13sin3t)y''+2y'+y=75\left(\sin t -\dfrac{1}{2}\sin 2t+\dfrac{1}{3}\sin 3t \right)

First, we will solve the corresponding homogeneous equation


Because this is a linear homogeneous differential equation with constant coefficients let

y=eλtyeλt,y=λ2eλty=e^{\lambda t}\Rightarrow y'e^{\lambda t}, y''=\lambda^2e^{\lambda t}

If we use the previous substitution in our homogeneous equation, we get:

(λ2+2λ+1)eλt=0λ2+2λ+1=0λ=1(\lambda^2+2\lambda+1)e^{\lambda t}=0\Rightarrow \lambda^2+2\lambda+1=0\Rightarrow \lambda=-1

Because we got double real roots, its solution will be:

yh(t)=et(c1+c2t)y_{h}(t)=e^{-t}(c_1+c_2 t)

We already know that the general solution of the given equation will be the sum of a general solution yhy_h of the homogeneous equation and any solution ypy_p.

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