Question

Show that the following initial value problem has a unique solution that exists for all of t

x'' + sin(x')cos(xt) = cos(x)cos(2t)

x(0) = 1, x'(0) = 0

Answer 1

x'' + sin(x')cos(x*t) = cos(x)*cos(2*t)

x(0) = 1, x'(0) = 0

we know if p(t), q(t), and g(t) be continuous on [a,b], then the differential equation

        y'' + p(t)y' + q(t)y = g(t)           y(t₀) = y₀     y'(t₀) = y'₀

has a unique solution defined for all t in [a,b].

here g(t)=cos(x)*cos(2*t) is continuous function.

and sin(x')*cos(x*t) is also a continuous function so the following problem has unique solution.