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Differential equation with external force
Consider the differential equation
$$x'(t)=y(t),$$
where $x(t)$ is a vector-valued function and $y(t)$ is a scalar valued function. Assume $x(t)$ and $y(t)$ are (solutions of) a dynamical system (which can be easily obtained by Euler or Runge-Kutta methods) and that the solution has reached a steady state (at time $t$).
Now, suppose we suddenly apply an external forcing $F(t)$ to the solution which increases linearly in time, from zero to a very large value. We can view the equation above as a dynamical system where the forcing $F(t)$ acts as an additional input (it is not part of the equations of the dynamical system).
Now, this modified equation is actually a full differential equation which we can solve. What I am struggling with is how to solve a differential equation with the new forcing and how to interpret the solutions. My first thought is that if we set $x'(t)$ from the original equation and $F(t)$ as coefficients, then we can compute the solution as $x(t)+F(t)$ (i.e. if we are solving for the $x(t)$’s solution in the original equation, then we can just add in the external force). This means that the forcing only affects the first order part of the solution (i.e. everything outside the brackets) but does not affect the evolution of the solution itself. Is this correct and/or is there a better approach?
A:
You cannot use $x'(t)=y(t)$ (or any other ODE) for describing your situation because the forcing $F(t)$ acts on $x(t)$ and not on $x'(t)$.
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