I tried to solve the following system of equations:
\begin{align*} x'(t) &= \frac{a \, |y(t)-x(t)|}{e^{\, b \, |y(t)-x(t)|}} - \frac{c}{|y(t)-x(t)|}\\ y'(t) &= - \, x'(t) \end{align*}
with initial conditions $x(0) = x_0$ and $y(0) = y_0$. There is one more constraint, namely $x(t) \neq y(t)$. Unfortunately, Mathematica calculated solutions for $x(t)$ and $y(t)$ which I am not familiar with. Furthermore, I did not manage to plot these solutions together with their corresponding vector field.
Here is my trial:
Eqs1:={P1'[t]==a*(Q1[t]-P1[t])*E^(-b*(Q1[t]-P1[t])),Q1'[t]==-P1'[t]};
Sols1 = DSolve[Eqs1,{P1,Q1},t]
Eqs2:={P2'[t]==-c/(Q2[t]-P2[t]),Q2'[t]==-P2'[t]};
Sols2 = DSolve[Eqs2,{P2,Q2},t]
Manipulate[Plot[{C[1]-InverseFunction[1/2 ExpIntegralEi[-b (-C[1]+2 #1)]&][-a t+C[2]]+C[1]+1/2 (-C[1]-Sqrt[4 c t+C[1]^2+4 C[2]]),InverseFunction[1/2 ExpIntegralEi[-b (-C[1]+2 #1)]&][-a t+C[2]]+1/2 (C[1]+Sqrt[4 c t+C[1]^2+4 C[2]])},{t,0,10},PlotRange->Automatic,ImageSize->Large],{a,0,10},{b,-5,5},{c,0,10},{C[1],0,10},{C[2],0,10}]
Basically, I tried solving the system by splitting the equations.
Is anyone able to help me out of this?