@acer you are right; I posted this question in December, and even you response to my question. Later, when I replied to your response, my reply was automatically deleted, and even several times I followed up on my qurey to to my question, but my reply was automatically deleted after some time. That is why I edited my question yesterday. 

My system does not have eta and I want to draw a stream line between y and x. Sir I have deleted x=0 in the file you uploaded above, and after that, my file is not producing results. Besides that, the expected result picture was uploaded to the question. Can you please help with this regard?

A few days ago, @acer helped me regarding my problem, but the replay has been deleted.

My system contains x and y; do not depend on eta. Regrading x=0, if I do not assign it to zero, do not get the results. Otherwise, there is no need to put x=0 because I am interested in plotting streamplots between y and x (y on the vertical axis and x on the horizontal axis). Besides this, I solved this system analytically, then considered the stream function, did some steps, and plotted the stream line. It is different from the stream function, which has been obtained directly by using the numeric method. I have assigned the values to parameters that I used during the analytical plot. I put x=0, and did not get the answer. Besides,  I am uploading the graph as a reference, which I have obtained by considering the stream function. This plot is similar to my flow direction, and I expect the same results from the numeric method..

streamline_Help_ac.mw

@Hullzie16 I have added the ODEs

Help_solution.mw

@acer Is it possible to check the residual error at once for the above file that you corrected... There are nine results and I am checking one by one by putting

evalf(eval(F, x = 0.009501474277))

@acer Thanks, sir, i was doing a mistake while use unapply to expression F.

@mmcdara Thanks, sir, for the help. Can you please explain a little bit for better understanding. Why did you calculate the minimum or maximum? or what is purpose of this line, hi := plottools:-transform((x, y) -> [x, y, minimum*1.2]):

@mmcdara I am increasing entries from 6 to 9, but I do not understand what is wrong with my file. Even your file is working properly.

my_try.mw

@mmcdara , When I increasing my entry means from 6 to 9, I got `[Length of output exceeds limit of 1000000], How can we deal with or fix this issue?

@mmcdara Thanks, sir, it is working accordingly. In some values like epsilon=[0.5$6] you put 6 at the end. What does that mean?

@mmcdara the answers are not same.

@C_R actually, h=1+x^2, and I am interested to computing the relation for x in terms of lambda. If explaicit solution is not possible, then is it possible to generate another expression in terms of lambda for x by taking random values of involved constants, like Br=0.2, m=1 etc.? by generatung data first and then make expression

@C_R , I need to separate the value of h as an expression in terms of other parameters. I do not want to give or suppose any numeric values to the independents, like Br, N, beta, etc

@mmcdara Thanks, sir. I am changing the initial gues in the Newton method, like x = 0.1, 0.8, 1, etc., every time the method gives me different answers. How can we assure that our roots are correct? I am thinking the answer should be the same whether you apply bisection, fsolve, or the Newton method to the given expression. Besides this, I am trying to apply a Newton procedure to my expression but have not yet gotten the answers. Can you please help in this regard as well?

Newton_Method_Help.mw

@acer thanks sir, its working perfectly