@Preben Alsholm 

what you say about nonlinear aigenvalue problem is true, I really forget this issue and substitude an arbitrary one.

according your guidance, can I use maple for finding eigenvalues of nonlinear equations? because I cant use an additional boundary condition for determining one according youe explanation.

I already has linear equations therefore I use eigenvalue command in maple easily for determining this.

Inaddition I really thank you for your help, your explanation is really complete and helpful.

THANKS ALOT.

If this is necessory I put my equations with constant parameter (P) which should be considerd in eigenvalue problem.

dsys3 := {-901.7765286*(diff(f1(x), x, x))+3.153274902*10^(-10)*(diff(f2(x), x, x, x))+0.4807756008e-5*(diff(f2(x), x))-27.92641797*(diff(f3(x), x))+18074.25056*f1(x)-3.100642108*(diff(f3(x), x, x))*(diff(f3(x), x))+7.74301535*(diff(f3(x), x))*f3(x) = 0, 0.1873004296e-2*(diff(f3(x), x, x, x, x))+(diff(f3(x), x, x))*(-0.1142850079e-2-0.1587291776e-1*P)-.1797931318*(diff(f3(x), x, x))+13.96314014*(diff(f1(x), x))+1.822500000*10^(-10)*(diff(f2(x), x, x))-4.888800000*10^(-7)*f2(x)+22.72337812*f3(x) = 0, 0.2249991345e-1*(diff(f2(x), x, x, x, x))-227.1861332*(diff(f2(x), x, x))+29537.78948*f2(x)-2.841067252*10^(-11)*(diff(f1(x), x, x, x))-3.39267254*10^(-7)*(diff(f1(x), x))+7.650000000*10^(-11)*(diff(f3(x), x, x))-1.400000000*10^(-7)*f3(x)-5.500000000*10^(-7)*f3(x)^2+9.278333333*10^(-13)*(diff(f3(x), x))^2+3.300000000*10^(-13)*(diff(f3(x), x, x))*f3(x) = 0, f1(0) = 0, f1(1) = 0, f2(0) = 0, f2(1) = 0, f3(0) = 0, f3(1) = 0, (D(f2))(0) = 0, (D(f2))(1) = 0, (D(f3))(0) = 0, (D(f3))(1) = 0}

@Preben Alsholm 

I understand what you say is true and interstingly f2(x) can be zero.

Thanks alot

For obtaining f1(x) and f3(x), I should follow according to your previous comments?

@Preben Alsholm 

Thank you for your attention.

but as you say I am looking for a nontrival solution.

@Preben Alsholm

dsys3 := {0.1873004296e-2*(diff(f3(x), x, x, x, x))-.3714109950*(diff(f3(x), x, x))+13.96314014*(diff(f1(x), x))+1.822500000*10^(-10)*(diff(f2(x), x, x))-4.888800000*10^(-7)*f2(x)+22.72337812*f3(x) = 0, -901.7765285*(diff(f1(x), x, x))+3.153274902*10^(-10)*(diff(f2(x), x, x, x))+0.4807756008e-5*(diff(f2(x), x))-27.92641797*(diff(f3(x), x))+18074.25057*f1(x)-3.100642108*(diff(f3(x), x, x))*(diff(f3(x), x))+7.74301535*(diff(f3(x), x))*f3(x) = 0, 0.2249991345e-1*(diff(f2(x), x, x, x, x))-227.1861332*(diff(f2(x), x, x))+29537.78948*f2(x)-2.841067252*10^(-11)*(diff(f1(x), x, x, x))-3.392672540*10^(-7)*(diff(f1(x), x))+7.650000000*10^(-11)*(diff(f3(x), x, x))-1.400000000*10^(-7)*f3(x)-5.500000000*10^(-7)*f3(x)^2+9.278333333*10^(-13)*(diff(f3(x), x))^2+3.300000000*10^(-13)*(diff(f3(x), x, x))*f3(x) = 0, f1(0) = 0, f1(1) = 0, f2(0) = 0, f2(1) = 0, f3(0) = 0, f3(1) = 0, (D(f2))(0) = 0, (D(f2))(1) = 0, (D(f3))(0) = 0, (D(f3))(1) = 0}

Thank you.