Plot a differential equation of two dependent variables?

Questions
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1

I would like to know how to plot

\frac{x(t)-X(t)}{y(t)-Y(t)} = \frac{y'(t)}{x'(t)}

X(t) and Y(t) are defined, such as e^t, Sin(t), etc., but I don’t know how to get Mathematica to plot this. I’ve given it proper initial conditions that anyone could start the DE by hand and draw out the answer, but don’t know how to do it for Mathematica.

2

had to add an extra condition to get as many equations as variables

In[13]:= sln = 
 NDSolveValue[{(x[t] - Sin[t])/(y[t] - Exp[t]) == y'[t]/x'[t], 
   x'[t]^2 + y'[t]^2 == 1, x[0] == 1, y[0] == 2}, {x[t], y[t]}, {t, 0,
    2}]

During evaluation of In[13]:= NDSolveValue::ndsvb: There are multiple solution branches for the equations, but NDSolveValue will return only one. Use NDSolve to get all of the solution branches.

ParametricPlot[sln, {t, 0, 2}]

deplot.jpg
deplot.jpg289×432 6.65 KB

3

Thank you!! I couldn’t seem to figure out how to write it properly!