# NDSolve: evolutionary singularity in coupled differential equations

I am trying to solve coupled differential equation where a function attains a value such that its derivative is infinite.

NDSolve[
{(B[x] x - (2/3) ft x^(5/2)) D[A1[x], x] + x A1[x] D[B[x], x] + A1[x] B[x] == 0,
(B[x] x - (2/3) ft x^(5/2)) D[B[x], x] - B[x]^2/2 + (fk (x D[A1[x]^γ, x] + A1[x]^γ))/A1[x] + 1 == 0,
A1[1] == a0, B[1] == b0},
{A1[x], B[x]}, {x, 0.001, 1}][[1]]


where a0 = 1, b0 = 1, ft = 0.0001, fk = 0.001 for $$\gamma > 1$$. The solution reaches a singular point when B[x] x - (2/3) ft x^(5/2) == 0.

Can you help me on how to solve this singularity problem?

• One of the singularities is obviously at x == 0. For gamma equal to 3/2, another is around x == 2. Is that what you're after? – Michael E2 May 27 at 15:35
• Basically I am looking on how to bypass when B[x] x - (2/3) ft x^(5/2) == 0. Do you know any method/technique for this? – mageshwaran T May 27 at 19:17
• Normally, one doesn't, I think. However, I did it here , which refers to another answer where I used it. Your problem looks harder, but maybe it's manageable. – Michael E2 May 27 at 19:51