Numerical solution of ODE

Question

I have two coupled non linear second order differential equation.

{
 -y1''[x] == -Exp[k1 (y1[x] - y2[x])] + Exp[-k2 (y1[x] - y2[x])],
 -y2''[x] == Exp[k1 (y1[x] - y2[x])] - Exp[-k2 (y1[x] - y2[x])]
}

How can I solve this numerically to get the values of y1 and y2 at different points using NDSolve.

Answer 1

Initial/boundary conditions are missing along with the values for k1 and k2. Therefore, I choose random ones.

Eq1 = -y1''[x] == -Exp[k1 (y1[x] - y2[x])] + Exp[-k2 (y1[x] - y2[x])]

Eq2 = -y2''[x] == Exp[k1 (y1[x] - y2[x])] - Exp[-k2 (y1[x] - y2[x])]

k1 = 1; k2 = 1;

sol = NDSolve[{Eq1, Eq2, y1[0] == 1, y1'[0] == 0, y2[0] == 1, y2'[0] == 0}, {y1, y2}, {x, 0, 10}]

xy = Table[{x, y1[x], y2[x]} /. First@sol, {x, 0, 10, 2}];

% // TableForm

Answer 2

I recommend you take a look at ParametricNDSolveValue:

{xmin, xmax} = {0, 1};
sol = ParametricNDSolveValue[
  {
   -y1''[x] == -Exp[kk1 (y1[x] - y2[x])] + 
     Exp[-kk2 (y1[x] - y2[x])], -y2''[x] == 
    Exp[kk1 (y1[x] - y2[x])] - Exp[-kk2 (y1[x] - y2[x])],
   y1[xmin] == yy10,
   y1[xmax] == yy11,
   y2[xmin] == yy20,
   y2[xmax] == yy21
  },
  {y1, y2},
  {x, xmin, xmax},
  {kk1, kk2, yy10, yy11, yy20, yy21}
]


Manipulate[
  Plot[
    Evaluate[Through[sol[k1, k2, y10, y11, y20, y21][x]]],
    {x, xmin, xmax}, AxesOrigin -> {0, 0}, PlotRange -> All
  ],
  {{k1, 1}, -1, 1},
  {k2, -1, 1},
  {{y10, 0}, -1, 1},
  {y11, -1, 1},
  {{y20, 0}, -1, 1},
  {y21, -1, 1}
]

If you need numeric values for a particular set of x-values, you can get them like this:

xValues = Range[0, 1, 0.1]
yValues = Through[sol[1, 1, 0, 1, 1, 0][xValues]]