# Variable coefficient Differential Equation

Can anyone help with the following differential equation. The errors i am getting are also given.

DSolve[{x y''[x] -  h*x  y'[x] + y'[x] == 0, y[0] == 0,
y[1] == 1}, y[x], x]


During evaluation of In[20]:= Solve::incnst: Inconsistent or redundant transcendental equation. After reduction, the bad equation is -C[1] == 0. >>

During evaluation of In[20]:= Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. >>

During evaluation of In[20]:= DSolve::bvnul: For some branches of the general solution, the given boundary conditions lead to an empty solution. >>

Out[20]= {}

• What is h, a function of x or a constant, if so is it real, positive or...? The equation is x y''[x] - h*x y'[x] + y'[x] == 0 or x y''[x] - h*x y'[x] + y[x] == 0? – Artes Mar 25 '16 at 14:15
• All too often, DSolve fails due to boundary conditions. This is the case here. – bbgodfrey Mar 25 '16 at 14:21
• h is a real positive constant – Asim Aziz Mar 25 '16 at 14:24
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## 1 Answer

y[0] == 0 is at fault here. Omit it to obtain

DSolve[{x y''[x] - h*x y'[x] + y'[x] == 0, y[1] == 1}, y[x], x][[1, 1]]
(* y[x] -> 1 - C[1] ExpIntegralEi[h] + C[1] ExpIntegralEi[h x] *)


However, ExpIntegralEi[h x] evaluates to Infinity at x = 0. Thus, y[0] == 0 has no solution.

y[x] /. % /. x -> 0
(* 1 + C[1] (-∞) - C[1] ExpIntegralEi[h] *)