91Ó°ÊÓ

Decide whether the method of undetermined coefficients together with superposition can be applied to find a particular solution of the given equation. Do not solve the equation.$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{6}\mathbf{y}\mathbf{\text{'}}\mathbf{-}\mathbf{4}\mathbf{y}\mathbf{=}\mathbf{4}\mathbf{sin}\mathbf{3}\mathbf{t}\mathbf{-}{\mathbf{t}}^2{\mathbf{e}}^{3t}\mathbf{+}\frac{1}{t}$

Question: find a general solution to the given differential equation.

$\mathbf{4}\mathbf{w}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{20}\mathbf{w}\mathbf{\text{'}}\mathbf{+}\mathbf{25}\mathbf{w}\mathbf{=}\mathbf{0}$

Find a particular solution to the differential equation. $\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\left(\mathbf{x}\right)\mathbf{+}\mathbf{y}\left(\mathbf{x}\right)\mathbf{=}{\mathbf{2}}^x$

In Problems 9 through 14, find a general solution to the given Cauchy–Euler equation for t>0.

${\mathbf{t}}^2\frac{{\mathbf{d}}^2\mathbf{z}}{{\mathbf{dt}}^2}\mathbf{+}\mathbf{5}\mathbf{t}\frac{\mathrm{dz}}{\mathrm{dt}}\mathbf{+}\mathbf{4}\mathbf{z}\mathbf{=}\mathbf{0}$

In Problems 11–18, find a general solution to the differential equation.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{y}\mathbf{=}\mathbf{tant}\mathbf{+}{\mathbf{e}}^{\mathbf{3}\mathbf{t}}\mathbf{-}\mathbf{1}$

Use the mass-spring analogy to explain the qualitative nature of the solutions to the Rayleigh equation (22) $\mathrm{y}-\left[1-{\left(\mathrm{y}\text{'}\right)}^2\right]\mathrm{y}\text{'}+\mathrm{y}=0$depicted in Figures 4.24 and 4.25.

A 1 – kg mass is attached to a spring with a stiffness of 100 N/m. The damping constant for the system is 0.2 N-sec/m. If the mass is pushed rightward from the equilibrium position with the velocity of 1 m/sec,when will it attain its maximum displacement to the right?

A mass weighing 8 lb is attached to a spring hanging from the ceiling and comes to rest at its equilibrium position. At t = 0, an external force F(t) = 2 cos 2t lb is applied to the system. If the spring constant is 10 lb/ft and the damping constant is 1 lb-sec/ft, find the equation of motion of the mass. What is the resonance frequency for the system?

Find a general solution to the given differential equation. ${\mathbf{t}}^{\mathbf{2}}\mathbf{x}\mathbf{\text{'}}\mathbf{\text{'}}\left(\mathbf{t}\right)\mathbf{+}\mathbf{5}\mathbf{x}\left(\mathbf{t}\right)\mathbf{=}\mathbf{0}\mathbf{,}     \mathbf{t}\mathbf{>}\mathbf{0}$

Find a particular solution to the differential equation.

$\mathbf{2}\mathbf{x}\mathbf{\text{'}}\mathbf{+}\mathbf{x}\mathbf{=}\mathbf{3}{\mathbf{t}}^2$