## Related questions with answers

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 $y-y_p$ to see when the system practically reaches the steady state.

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

Solution

VerifiedWe are actually given the following equation:

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

First, we will solve the corresponding homogeneous equation

$y''+2y'+y=0$

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

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

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

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

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

$y_{h}(t)=e^{-t}(c_1+c_2 t)$

We already know that the general solution of the given equation will be the sum of a general solution $y_h$ of the homogeneous equation and any solution $y_p$.

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