I have an equation: $ $ y[x]=\dfrac{b}{\sqrt{|k(x)|}}e^{-\int^x k(x’)dx’}\qquad\text{(1)}$ $

Where

$ $ k(x)=\sqrt{\dfrac{w^2}{c^2}+\left(\dfrac{w(w_p)^2}{c^2v_t h}\right)(A1)}$ $

$ $ A1=\dfrac{1}{\sqrt{\pi}}\int\dfrac{e^{-y^2}}{y-\dfrac{w+2iR-om}{v_t(\dfrac{w}{c}(1-\dfrac{(1-\dfrac{x}{L})}{1+\dfrac{w_c}{w}})^{\dfrac{1}{2}})}}dy$ $

This expression $ \int^x \sqrt{\dfrac{w^2}{c^2}+\left(\dfrac{w(w_p(x’))^2}{c^2v_t h(x’)}\right)A1(x’)}dx’$ … (2) means that:

At first I must solve $ \int^x \sqrt{\dfrac{w^2}{c^2}+\left(\dfrac{w(w_p(x’))^2}{c^2v_t h(x’)}\right)A1(x’)}dx’$ , then change $ x$ to $ x’$ .

But it has an error. Why? Because I get $ A1(x’)$ numerically and put its answer to (2), so it will be mistake. In fact I can’t solve the indefinite integrate (2) because $ A1$ is solved numerically, so the error is right.

The plot of the equation (1) muse be as below:

The condition is $ y[-L]=10$ or $ y[-L]=0$ and $ L=0.00001$ . the others show in the program.

R = ((1 - x/L)*W^2*(16*10^-20)^2*Pi*Sqrt[me]*LnLumbda)/( T^(3/2)*e0);  Simplify[   1/Sqrt[π] E^-S^2/(S - (W + 2 I R - Om)/     (vt (W/CC (1 - (1 - x/L)/(1 + (Wc)/W))^(1/2))))] 

-((0.5641895835477563 E^-S^2 Sqrt[x]) /   ((0.016006834477809786 + 0.00008178287140291831 I) - 1. S Sqrt[x] - (0. + 8.178287140291829 I) x)) 

A1[x_] :=    NIntegrate[     -((0.5641895835477563 E^-S^2 Sqrt[x])/       ((0.016006834477809786 + 0.00008178287140291831 I) - 1. S Sqrt[x] -          (0. + 8.178287140291829 I) x)),      {S, -∞, ∞}]  yWKB[x_] := b/Sqrt[Abs[κ[x]]] Exp[NIntegrate[κ[xp], {xp}]]  κ[x_] := Sqrt[W^2/CC^2 + (W wp^2)/(CC^2 vt k) A1[x]]  Simplify[κ[x]] 

Sqrt[(3.947841760435743`*^13 Sqrt[   x] + (6.319244960388013`*^11 -      6.319244960388012`*^16 x) NIntegrate[-((     0.5641895835477563` E^-S^2 Sqrt[      x])/((0.016006834477809786` + 0.00008178287140291831` I) -       1.` S Sqrt[       x] - (0.` +          8.178287140291829` I) x)), {S, -∞, ∞}])/Sqrt[x]] κ[x_] := √(3.947841760435743`*^13 +      1/Sqrt[x] (6.319244960388013`*^11 -         6.319244960388012`*^16 x) NIntegrate[-((0.5641895835477563` \ E^-S^2 Sqrt[            x])/((0.016006834477809786` + 0.00008178287140291831` I) -             1.` S Sqrt[             x] - (0.` +                8.178287140291829` I) x)), {S, -∞, ∞}]) 

yWKB[-L] 

NIntegrate::ilim

0.0004027134076113933` b E^NIntegrate[κ[xp], {xp}] 

Solve[0.0004027134076113933 b E^NIntegrate[κ[xp], {xp}] == 10, b] 

NIntegrate::ilim

{{b -> 24831.554676346186` E^(-1.` NIntegrate[κ[xp], {xp}])}} 

Plot[Abs[yWKB[x]], {x, -L, L}] 

So how do I solve y[x]?