@Kitonum thanks sir for your help,  can you please correct my worksheet attached below. The same thing you did above for simple equation, I applied for complex equation but could not get the answer. if it is possible to solve numerically the how could a obtain the expression for lambda?

help_ode_solution.mw

@Mariusz Iwaniuk  thanks sir, I am not well familiar with maple use. can you correct my worksheet attached below. The same thing you did above I applied for complex equation but could not get the answer

help_ode_solution.mw

@Mariusz Iwaniuk  thanks sir, why you put this 6*(h - 1)*(h - k)/(h^2*(-k^2 + 1)) in the last line, and what is the meaning of %

@Kitonum bundle of thanks sir by the way why we take eval(rhs(%), x = 1);? rhs stand for right-hand side or is there any other meaning and % meaning?

@Kitonum sir i am looking for how to find the value of lambda?

@Kitonum why it gives error

restart;
h := k - (k - 1)*x;
DE := diff(p(x), x) = 6/h^2 - 12*lambda/h^3;
             d                6              12 lambda    
      DE := --- p(x) = ---------------- - ----------------
             dx                       2                  3
                       (k - (k - 1) x)    (k - (k - 1) x) 
BC := p(0) = 0, p(1) = 0;
                    BC := p(0) = 0, p(1) = 0
simplify(dsolve({BC, DE}, p(x)));
Error, invalid input: simplify uses a 1st argument, s, which is missing

@Kitonum thanks sir, it is working well, i have confusion in the last step can you please explain how and why you did this

p:=unapply(eval(p(x),%), x);
h:='h':
eval(p(x),x=solve(h=k-(k-1)*x,x));
simplify(%);

and how to fine the value of lambda=k/1+k after integration

@Kitonum Thanks sir for your effort but i am looking for the solution which shows in the attached picture. first, need to put the value of h in the given first order ode and then solve obtained expression by using BCS that you b the same as p in the picture and neet to put the expression p=0 solve for lambda. but I do not obtained the exact value of p that is given papers

@Mariusz Iwaniuk pls have a look and required results

@Mariusz Iwaniuk that better idea can you please update the sheet that how could I generate the different graph in the same picture for different value of M=0.1----0.9 and for fix k=2 etc

@Mariusz Iwaniuk can you please give a little explanation that you did in the sheet and why you take rhs(sol)?

@Carl Love I have already solved this by using dsolve command now I am looking for the algorithm of FDM

@acer Thanks sir, it is working and your way of helping is always helpful for me. But i don't understand why you convert into EXP?

@vv sorry sir in the first file there was a mistake in expression. Still, at my pc, it is not giving result and not integrating, I do not know why?

Help.mw

@vv it is not working, can you please have a look on my try, pls have attached work

Help.mw