Question

a) The homogeneous and particular solutions of the differential equation ay'' + by' + cy = f(x) are, respectively, C1exp(x)+C2exp(-x) and 3x^3. Give the complete solution y(x) of the differential equation.

b) If the force f(x) in the equation given in a) is instead f(x) = f1(x) + f2(x) + f3(x), where f1(x), f2(x), and f3(x) are generic forces, what would be the particular solution?

c) The homogeneous solution of a forced oscillator is cos(t) + sin(t), what is the friction coefficient of the system?

d) If the mass associated to the harmonic oscillator in c) is m=1, what is the value of the elastic constant?

e) If a force F(t) = 5sin(t) is applied to the harmonic oscillator in c), could the particular solution be Asin(t)?

Answer 1

(a) The complete solution is just the sum

(b) In this case 3 particular solutions may be found. They all should be added up to get the full particular solution.

(c) There is no decaying factor in the homogeneous part so there can't be any friction. The solution just corresponds to a SHM whose differential equation is

(d) The angular frequency is given by the coefficient of time, i.e.,

Thus, if m=1, k=1.

(e) The governing equation is

Clearly can't satisfy the above equation because LHS becomes 0 no matter what is A.