@sharena2 

I think it should work in Maple 16, 17, 18

There were many things missing. anYWAY, check the attached sheet which is working fine.

@Carl Love Thanks dear. The approach suggested by Preben Alsholm is more efficient.

I have just one question, for

H2:=(((((((1+lambda*m*x/a0+phi*((8/Pi^3*Sum((-1)^(n+1)/(2*n-1)*sin(2*Pi*(2*n-1)*x),n=1..infinity)))))))))); 

S2:=8/Pi^3*Sum((-1)^(n+1)/(2*n-1)*sin(2*Pi*(2*n-1)*x),n=1..infinity);

can I define the function 

g1:=piecewise(x<1/8,1,x<3/8,-1,1);

Cheers!

@Carl Love please see the attached sheet. Its taking long time to plot. sheetplot1.mw

@Markiyan Hirnyk Thanks for your kind help. You are right, in this case we have non-trivial sol.

Sorry for not appreciating your help on time.

@Carl Love It works but take a little longer to excute the plot.

If you only change k=0.9 then you will understand my point?

@Markiyan Hirnyk Thx for your efforts

@Markiyan Hirnyk 

404 - File or directory not found.

@Carl Love 

plot([subs(tau=0.1,DP3),subs(tau=0.2,DP3)], 0 .. 1, axes = box, linestyle = 1, color = [red,green], numpoints = 30);

@Markiyan Hirnyk Thanks, it worked but here is another problem, I want to have multiline/values lineplot, where I can so-called vary a parameter, in this case tau.

restart:with(plots):
E:=0.2:phi:=0.2:alpha:=0.1:k:=1:lambda:=0.1:a0:=0.5:m:=0.1:
f:=sin(alpha)/E:
h3:=(evalf(1+lambda*m*x/a0+phi*Re((4/Pi*sum((-1)^(n+1)/(2*n-1)*cos(2*Pi*(2*n-1)*x),n=1..infinity))))):
g3 := unapply(f-(q*(k+1)*(k+2)/((1-tau)^(k+1)*h3^(k+2)*(k+1+tau)))^(1/k), x, Q,tau):
DP3 := proc (Q,tau) options operator, arrow; int(evalc(Re(g3(x, Q,tau))), x = 0 .. 1, numeric) end proc:
plot([subs(tau=0.1,DP3),subs(tau=0.2,DP3)], 0 .. 1, axes = box, linestyle = 1, color = [red,green], numpoints = 30);

Warning, unable to evaluate the functions to numeric values

@ANANDMUNAGALA 

I am glad that somehow you got some help from MP but I am still wondering about your exact Q.

@ANANDMUNAGALA 

paste your try here

@Preben Alsholm Thanks for your help

@Preben Alsholm 

uu:=rhs(dsolve({ODE,bcs}));

U:=(evalf(uu)); # using the value of m here

@Preben Alsholm Is it possible to write a proc  which could hold fsolve (finding m ) while plotting for  different values of a parameter,  how to got the following plot?

plot([subs(phi=0.0,U),subs(phi=0.2,U)],y=-1..1,axes=box,color=[red,green],adaptive= false);