Finding the differential equation (initial value problem)

[djh101]

Homework problem for nonlinear dynamics.


Let us write x_λ(t) for the solution of the initial value problem

\dot{x} = f(x) & x(0) = λ​

where f is continuously differentiable on the whole line and f(0) = 0.
a) Find the differential equation for \frac{∂x_{λ}}{∂λ}(t)


I'm a little confused about where to begin, so any sort of push in the right direction would be appreciated. Also, what does the λ subscript mean?

 

[tiny-tim]

hi djh101!

\dot{x} = f(x) & x(0) = λ​

…
I'm a little confused about where to begin, so any sort of push in the right direction would be appreciated. Also, what does the λ subscript mean?

x_λ(t) is the solution to dx/dt = f(x) with initial condition x(0) = λ

hint: what can you say about x_λ+dλ(t) - x_λ(t) ?

 

[djh101]

x_λ+dλ - x_λ = x_dλ = ∂x_λ?