Question: On solving differential equation

Dear All,

I have the following differential equation:

(diff(lambda(t), t) = lambda(t)*(rho - A*beta*h^(1 - beta + a)/(A*h^(1 - beta + a)*t*(1 - beta) + k^(1 - beta))))     (1)

where rho, A, beta, a, h and k, are known parameters. Note, that all of them are positive and parameters rho, beta and a are less than one.

When solving this equation by dsolve, I get:

lambda(t) = _C1*exp(rho*t)*(A*h^(1 - beta + a)*t*beta - A*h^(1 - beta + a)*t - k^(1 - beta))^(beta/(-1 + beta))    (2)

which when substituting, for instance, beta=0.3 into the obtained solution, transforms into:

lambda(t) = _C1*exp(rho*t)/(-0.7*A*h^(0.7 + a)*t - k^0.7)^0.4285714286                                                                (3)

However, if I already have put beta=0.3 into the original differntial equation (see (1)) and solve it, I receive:

lambda(t) = _C1*exp(rho*t)/(10*k^(7/10) + 7*A*h^(7/10 + a)*t)^(3/7)                                                                       (4)

Clearly, the resultant solutions in (3) and (4) are quite different since the solution of (3) is any complex number and in (4) it is a real number. Note, that when solving the differential function (1) numerically, there exists a solution for any value of beta.

I also have tried to rename beta (supposing that maybe beta is any protected name) but the result remains the same. Also note, that if such manipulation is done with any other parameters, the solutions (3) and (4) are the same.

I will be thankful for any clarification regarding this issue. 

Thanks in advance,

Dmitry