I am attempting to numerically solve the following differential equation that includes differentiation with respect to two variables with two separate boundary conditions.
$y^{\prime\prime}(x) + \frac{3}{x}y^{\prime}(x) = \frac{dU}{dy}; y(\infty) = 0,\ y^{\prime}(0) = 0, U(y) = \frac{1}{4}y^{4}(\gamma +\alpha\ln^{2}y + \beta\ln^{4}y),$
where ${}^{\prime}$ indicates differentiation with respect to $x$.
Here is my attempt thus far is:
U[y_] = 1/4 y^{4}*({\[Gamma] + \[Alpha]*(Log[y/Mp])^{2} + \[Beta]*(Log[y/Mp])^{4}});
eqn = D[y[x], {x, 2}] + 3 D[y[x], x]/x - U'[y[x]] == 0;
NDSolve[{eqn, y[100] == 0, y'[0.1] == 0}, y[x], {x, 0, 100}]
The value of the constants in the equation are as follows:
Mp = 2.435*10^18;
\[Alpha] = 1.4*10^-5;
\[Beta] = 6.3*10^-8;
\[Gamma] = -0.013;
\[Lambda]6 = 0;
But unfortunately I receive the following error messages:
Power: Infinite expression 1/0 encountered.
Any help on this matter would be greatly appreciated.
Update: I feel this might be possible to solve via a shooting method but I am unsure how to implement this.
U[y_]=
, and then write U'[y[x]] in eqn. That'll fix the first issue, but you still have a problem with U'(y) being undefined when y=0, which you have at the right boundary condition. $\endgroup$