Dear Dr. Rashed

You can see the following papers

www.damtp.cam.ac.uk/user/.../NA2013_09.pdf

www.sciencedirect.com/science/.../S0377042700004805

https://cs.uwaterloo.ca/research/tr/1994/03/ji03.pdf.

Professor Khater has some papers to solve

Sturm-Liouville problems

Some of my papers can found in my site

Dear Dr

Some of my papers

1- A. R. Seadawy, Fractional solitary wave solutions of the

nonlinear higher-order extended KdV equation in a stratified

 shear flow: Part I, Comp. and Math. Appl. \textbf{70} (2015)

 345–352.

2- Khater, A. H., Callebaut D. K., Malfliet, W. and Seadawy A. R., Nonlinear Dispersive

Rayleigh-Taylor Instabilities in Magnetohydro-dynamic Flows,

Physica Scripta, 64 (2001) 533-547.

3- Khater, A. H., Callebaut D.

K. and Seadawy A. R., "Nonlinear Dispersive Kelvin-Helmholtz

Instabilities in Magnetohydrodynamic Flows" Physica Scripta, 67

(2003) 340-349.

4- A.R. Seadawy, Exact solutions of a two-dimensional

nonlinear Schrodinger equation, Appl. Math. Lett. 25 (2012) 687.

5- A.R. Seadawy, Stability analysis for Zakharov-Kuznetsov equation of weakly

nonlinear ion-acoustic waves in a plasma, Computers and

Mathematics with Applications 67 (2014) 172-180.

6- A.R. Seadawy, Stability analysis for two-dimensional ion-acoustic waves in

quantum plasmas, PHYSICS OF PLASMAS 21 (2014) 052107.

7- Helal, M. A. and  Seadawy A. R., Variational method for the

derivative nonlinear Schrodinger equation with computational

applications, Physica Scripta, 80, (2009) 350-360.

8- Helal, M. A. and  Seadawy A. R., Exact

soliton solutions of an D-dimensional nonlinear Schrodinger

equation with damping and diffusive terms, Z. Angew. Math. Phys.

(ZAMP) 62 (2011), 839-847.

9-  Khater, A. H., Callebaut D. K. and Seadawy A. R., General

soliton solutions of an n-dimensional Complex Ginzburg-Landau

equation, Physica Scripta, Vol. 62 (2000) 353-357.

10- Khater, A. H., Helal M. A. and Seadawy A. R.,

General soliton solutions of n-dimensional nonlinear Schrodinger

equation" IL Nuovo Cimento 115B, (2000) 1303-1312.

11-  Khater, A. H., Callebaut D. K., Helal, M. A. and  Seadawy A.

R., Variational Method for the Nonlinear Dynamics of an Elliptic

Magnetic Stagnation Line, The European Physical Journal  D, 39,

(2006) 237-245.

12-  Khater, A. H., Callebaut D. K.,

Helal, M. A. and  Seadawy A. R., General Soliton Solutions for

Nonlinear Dispersive Waves in Convective Type Instabilities,

Physica Scripta, 74, (2006) 384.

13- Seadawy A. R., Three-dimensional nonlinear modified Zakharov–Kuznetsov

equation of ion-acoustic waves in a magnetized plasma, Computers

and Mathematics with Applications 71 (2016) 201-212.

14- M.A. Helal and A.R. Seadawy, Benjamin-Feir-instability in

nonlinear dispersive waves, Computers and Mathematics with

Applications 64 (2012) 3557-3568.

15- Seadawy, A.R., and El-Rashidy, K., Traveling wave solutions for some coupled nonlinear

evolution equations by using the direct algebraic method, Math.

and Comp. model. \textbf{57} (2013) 1371.

16- A. R. Seadawy, New exact solutions for the KdV equation with higher  order nonlinearity

by using the variational method, Comp. and Math. Appl. 62 (2011)

3741-3755.

17- Seadawy A. R, Stability analysis solutions for nonlinear

three-dimensional modified Korteweg-de Vries-Zakharov-Kuznetsov

equation in a magnetized electron-positron plasma" Physica A:

Statistical Mechanics and its Applications  Physica A 455 (2016)

44-51.

18- A.R. Seadawy, Nonlinear wave solutions of the three-dimensional

Zakharov–Kuznetsov–Burgers equation in dusty plasma, Physica A

439 (2015) 124–131.

19- A.R. Seadawy, Stability analysis of traveling wave

solutions for generalized coupled nonlinear KdV equations, Appl.

Math. Inf. Sci. 10 (1) (2016) 209–214.

For simplicity, put your ODE into first order form by letting x=[x_1 x_2]^T=[y y']^T, where ()^T means the transpose operator, and then writing your ODE in the form x'=f(x,t).  In your case, f=[x_2, (q(t)-\mu^2)x_1]^T.  Next, write an ODE system for the partial derivative with respect to some parameter p. Generally,

  (d/dt)(\partial x/\partial p) = (\partial/\partial p)(dx/dt) = (\partial f/\partial p) + (\partial f/\partial x)(\partial x/\partial p)

I'll leave it to you to take the partial derivatives of f.  If we rename z=\partial x/\partial p, then the equation above gives an ODE of the form z'=g(x,z,t).

You need an initial condition for z.  For example, in the case where p is \alpha_1, you have x(0)=[\alpha_1, 0]^T, so z(0)=(\partial x/\partial \alpha_1)(0) = [1, 0]^T. 

Since x appears in g(x,z,t), the ODE for z has to be solved simultaneously with the original ODE for x.  Numerically integrating the combined ODE system from t=0 to t=1 will give you x(1) and z(1)=(\partial x/\partial p)(1).

This basic approach applies to your Sturm-Louisville ODE and also applies more generally. 

I hope that helps you.

It seems, that You have initial value problem. So it have solution for any \mu. So You have only problem to solve numerically Your equation. I think it is not a problem. Then You can solve for 2 value of parameter and obtain derivative numerically (not good way). Or, solve  eqvation

-p''(t)+q(t)p(t)=\mu^{2}p(t),

p(0,\mu):=\partial_{\lambda} \alpha_{1}, p'(0,\mu):=\partial_{\lambda} \alpha_{2}

where p=\partial_{\lambda}y(1,n)

It is the same problem. If \mu depend from \lambda it will be little more complex, but not much. I think my suggestion is the same as Charles Wampler  suggested.

The eq -p''(t)+q(t)p(t)=\mu^{2}p(t) is not the best for numerical solution, but I think the Mathematica can do it easily in most case. Or You can use the Riccati equation method or filon method.

Thanks for all.  Dear  Konstantin V Koshel

How can I do that by Mathematica ??

Dear Rashad Mudhish Asharabi,

At first, You must understand how to solve initial value problem for ODE. Matematica is only the instrument. Next, You can see help for NDSolve. I recommend You test on the case with constant q, when You can solve problem analytically or by dsolve.

It is You problem in simplest case

-y''(t)+q_0(l)y(t)=\mu(l)^{2}y(t),

y(0,\mu):=\alpha(l)_{1}, y'(0,\mu):=-\alpha(l)_{2}

Let us try to find solution as exp(bt), then if we substitute it in equation, we have (l is the \lymbda, so we do not know what is the \lyambda)

b^2=q_0(l)-\mu(l)^{2}; b_1(l)=sqrt(q_0(l)-\mu(l)^{2})

or the solution is

y(t;l)=C_1(l)exp(b_1(l)t)+C_2(l)exp(-b_1(l)t)

You can define С_1,С_2 from initial condition. In that case You have analytical solution, as function from l and You can differentiate it.

Now You can use ndsolve for that problem, and verify if the numerical solution coincides  with analytical one.

Next You can use ndsolve for q(t,l). But be careful in that case.