2. As we will study in the coming weeks, the one-dimensional motion of a mass attached to aspring without friction is governed by the linear second-order equation with constant coeffi-cients(1)d²xdt²+w²x = 0,where w2 is a positive parameter.(a) Find the general solution to equation (1). You should have found a periodic solution:can you say what the period is?(b) Find the particular solution assuming that the mass is initially still in position x att = 0, i.e., using the initial conditions x(0) = ₁ and (0) = 0.(c) With friction, the motion is described by the equationd²xdxdt+w²x = 0,dt²where both f and w² are positive parameters. Find the general solution to equation (2) andsketch the behaviour of the solution in time (remember to discuss all the possible cases!). Isthis a periodic function? If so, can you find the period?+ f(If you're interested in the physical modelling details: equation (1) is the fundamental equa-tion of the simple harmonic motion; the parameter ² is a compound parameter defined asw² = k/m, where m is the mass of the particle and k a parameter characteristic of the spring.Equation (2) is the fundamental equation of the damped harmonic motion and f describesthe friction intensity. Later in the semester we will discuss these applications in more detail.) 2. As we will study in the coming weeks, the one-dimensional motion of a mass attached to aspring without friction is governed by the linear second-order equation with constant coeffi-cients(1)d²xdt²+w²x = 0,where w2 is a positive parameter.(a) Find the general solution to equation (1). You should have found a periodic solution:can you say what the period is?(b) Find the particular solution assuming that the mass is initially still in position x att = 0, i.e., using the initial conditions x(0) = ₁ and (0) = 0.(c) With friction, the motion is described by the equationd²xdxdt+w²x = 0,dt²where both f and w² are positive parameters. Find the general solution to equation (2) andsketch the behaviour of the solution in time (remember to discuss all the possible cases!). Isthis a periodic function? If so, can you find the period?+ f(If you're interested in the physical modelling details: equation (1) is the fundamental equa-tion of the simple harmonic motion; the parameter ² is a compound parameter defined asw² = k/m, where m is the mass of the particle and k a parameter characteristic of the spring.Equation (2) is the fundamental equation of the damped harmonic motion and f describesthe friction intensity. Later in the semester we will discuss these applications in more detail.)

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Accepted Answer

Part (a): General Solution without Friction[ x(t) = A\cos(wt) + B\sin(wt) \] Part (b): Particular…