For each of the equation in the following list determine the

For each of the equation in the following list, determine the set of initial conditions for which the Existence and Uniqueness Theorem (EUT) guarantees a unique solution:

(a) x\' = kx

(b) x\' = sinx

(c) x\' = t/x

(d) x\' = cost + x^3 x

Solution

x\'=kx

integrating we get f(x)=kx^2/2+c f(0) =c f\'(0)=0

x\' = sinx

integrating we get f(x) = -cosx+c both are continous from(-pi,pi)

x\'=t/x

integrating we get f(x) = t logx +c

x\' = cost + x^3 x

integrating we get f(x) =x^4/4x^2/2+cos(t)x+C