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Consider the equation $dx/dt = f(x, m)$, where $x$ is a function of $t$, x and $m$ is a parameter.

In Mathematica notation, f[x, m] is a function of x[t] and some parameter m.

Solutions of $dx/dt = f(x, m)$ vary with $t$ with the initial value $x(0)=x0$, and with the parameter $m$. When a certain $f(x, m)$ is given, how can I make a notebook that produces graphs of the solution sets of $dx/dt = f(x, m)$ along with controls to manipulate the values of $x0$ and $m$?

Examples: $f1(x) = x(m - x),\, f2(x) = 1 + m - x^2$

I can't really conceptualize solving this type of differential equation with Mathematica, where $x$ is a function of an independent variable. I have learned to solve ODE's with DSolve, but how does this function work if the variable is also a function of another variable?

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Can't you just use DSolveValue in the usual way (an example using your f1 function):

With[{expr = DSolveValue[{x'[t]==x[t](m-x[t]), x[0]==x0}, x[t], t]},
    Manipulate[Plot[expr, {t,0,1}], {{x0,1},0,10}, {{m,2},0,10}]
]

Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information.

enter image description here

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