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I have vector-valued parametric $C^2$ function $\{x(t),y(t): t \in [t^*,1]\}$ with $t^* \in (0,1].$ I know that $(x(t^*),y(t^*))=(x^*,y^*) \in R^2_+$ (+ probably irrelevant) and I know that $$\frac{x'(t)}{y'(t)}=-t$$ for all $t \in [t^*,1]$. What is the class of functions that satisfy those conditions?

My guess is that all solutions have the form, for all $t \in [t^*,1]$ and for an arbitrary scalar $k \neq 0$: $$x(t)=x^*+ k(t-t^*)$$ and $$y(t)=y^*+k [ln(t^*)-ln(t)].$$

But are there other solutions? And, if (bif if) I'm correct, how do I prove that there are not other solutions?

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Since you only have one equation, but two unknown functions you should expect to get a solution in terms of one "functional degree of freedom".

Indeed you can find solutions for arbitrary $x(t)$, (with $x(t^*)=x^*$). Just define $$ y(t)=y^*-\int_{t*}^tsx(s)ds $$ and you have a solution. (Of course this also works vice versa with arbitrary $y(t)$, which yields the same class of solutions but in a different form.)

Conclusion: The solutions you proposed in the question are only a subset of the whole class of solutions.

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