@tomleslie thank you. Our fluid is a non-newtonian fluid, so I think the value of Pr is less than 1. I think this is a higher version of the MAPLE code, which you have attached. I was checking if Pr>0.4 gives an error (why?). Is it the RKF45 method?.

@Preben Alsholm  @tomleslie    

Thank you so much, sir. I have added one equation of previous equations. Previous equations belonged to one of the papers. I want to solve this system of equations in the RK4 or RKF45 method by shooting technique. After that, I will learn how to solve it. My senior helped me to solve and plot this system of equations in MATLAB. But I have no knowledge of it. Kindly help to solve in MAPLE. See the attached worksheet problem.

help.mw

@tomleslie  and @Preben Alsholm Thank you both. But How to install the package of DirectSearch. Kindly suggest me.

@tomleslie  Which method you have used (attached) to solve this problem? Is this the RKF4 method or the RK shooting method?

@tomleslie  Thank you so much. Which version of MAPLE you are using? I am using MAPLE 2019. I think it is not working in my case. Did you solve it by direct method?

@Carl Love  Thank you. I have attached a pdf link please check this. I need this type of figure. lambda and alpha both are the same.

I need the result for alpha=1,2,3,4 by fix the values k=1.5. The Author has not written properly about infinity.  He has written that `" Although the problem is a boundary value problem, it is converted to an initial
value problem. We assign a trial value to f'(0) and integrate the equations using a Runge-Kutta algorithm and check whether the boundary condition at infinity is satisfied". Please help to find it.

Thank you

jpg2pdf.pdf

@tomleslie 

@nm ode.mw

I have attached please help to plot it. 

@Preben Alsholm 

@Carl Love 

@nm  yes there is a method sinh and cosh to find exact solutions.

@Mariusz Iwaniuk

 thank you, professor. I am trying to extend it.

@Mariusz Iwaniuk thank you, professor. I did not need Homotopy but now I got some knowledge about this. I will contact If I will work on Homotopy.

Thank you

@Mariusz Iwaniuk thank you. But I need its infinite series solution.

@Preben Alsholm But in the original equation have not boundary or initial conditions. How can I take arbitrary boundary conditions? It gives a solution of g(x). Will we solve the following nonlinear single ODE-

>ode := diff(f(x), x, x, x) = (-2*diff(f(x), x)*c^2*t^2 - 6*t*diff(f(x), x, x)*c^2*e^(c*x) - 6*diff(f(x), x)*f(x)*t^2 + diff(f(x), x)*t^2)/(2*c^2*(e^(c*x))^2)