Let $$\frac{dx}{dt}=(x-1)(x+4)\sin(t)+(x-1)$$ be be a non-linear non-separable first order differential equation.
(a) Verify existence and uniqueness is satisfied for all $x$ and $t$.
(b) Verify $x(t) = 1$ for all $t$ is a solution if $x(t_0) = 1$.
(c) Explain why any solution to this differential equation is either always one or never one.
I got part a) Since $f(t,x)$ and $\frac{df}{dx}$ are both continuous for all x and t they satisfy the Existence Uniqueness Theorem.
How do I go about part b), do I just take $x(t)=1$, find its derivative and check if it works?
Also can anyone explain part c).
Any help or hints will be appreciated.
Edit: After some though I think I figure out b), since the differential equation satisfies the Existence-Uniqueness Theorem for all x and t, then it must satisfy it for the initial condition of $x(t_0)=1$. This means that there is one and only one solution for this initial condition. Since $x(t)=1$ for all t including $t_0$, it would be sufficient to show that $x(t)=1$ is a solution to the DE to prove the statement. Is this right?
Best Answer
For b), yes, you only have to check the constant $1$ is a solution.
For c), you have to prove that if $x(t)$ can take the value $1$, it IS the constant function $1$ of b). Think of uniqueness.