# Getting a differential equation from a parametric function family

I have two parametric function families and i need to find the differential equation they satisfy.
I'm very lost on this one. I tried to write $$y=y(x)$$ using similarities between the two families.
But i couldn't come far as what i got was always far from the real solution. $$x=ce^{-t}-2t+2 \\\ y=c(1+t)e^{-t}-t^2+2$$ How sound i approach such problems?
Edit: I am looking at functions $$x$$ and $$y$$ as on two parametric function. So they are functions of parameter $$t$$. I want to find some interdependece between them and possibly some derivatives. Both $$x$$ and $$y$$ are dependant on $$C$$, which is a constant. The final equation should be without it.
Shortly, I am looking for a differential equation that those functions solve.
I tried with something like $$\frac{dy}{dx} = \frac{dy}{dt} \frac{dt}{dx}$$, but I didn't get any good results.

• First of all, the differential forms tag is inappropriate, as is the analysis tag. Your question is not clear. You have two functions of $t$, each depending on a parameter $c$? You're saying that these are both the general solution of a single (presumably second-order?) ordinary differential equation? Please make your question clearer. Oct 27, 2019 at 22:28

$$x=ce^{-t}-2t+2 \tag 1$$ $$y=c(1+t)e^{-t}-t^2+2 \tag 2$$
I agree with the Ted Shifrin's comment. Supposing that the problem is to eliminate $$c$$ and $$t$$ in order to get a differential equation involving only $$x,y,\frac{dy}{dx},\frac{d^2y}{dx^2},...$$ , they are an infinity of possible answers depending on the kind of differential equation and the order expected.
Take $$ce^{-t}$$ from $$(1)$$ and put it into $$(2)$$ $$y=(1+t)(x+2t-2)-t^2+2 \tag 3$$ Differentiate $$(1)$$ and $$(2)$$ $$\begin{cases} \frac{dx}{dt}=-ce^{-t}-2 \\ \frac{dy}{dt}=-ce^{-t}t-2t\end{cases}\quad\implies\quad \frac{dy}{dx}=t \tag 4$$ Putting $$t=y'=\frac{dy}{dx}$$ from $$(4)$$ into $$(3)$$ gives the ODE : $$y=(1+y')(x+2y'-2)-y'^2+2$$ But probably this is not what the question is asking for. The question should be clarified.