Deflection of Tapered Beam with Elliptic Cross Section

[Doha]

I am working on deriving expression for deflection of a tapered beam with an elliptic cross-section. Hence, area moment of inertia is a linear function of the beam length. The beam is fixed at one end, and a concentrated force F is applied on its tip at the free end. I am using the known relation between second derivative of deflection and moment. To verify my solution of the first integral (yielding the slope), I compare the result with the expression for the slope of tapered beam with circular cross-section under the same boundary conditions, by equating the minor and major semi-axes of the elliptic cross-section. The problem is: by equating the minor and major axes of the elliptic cross section, the solution diverges (resulting in 0/0). I tried integration manually and with mathematica, but the solution always diverges. Attached is my solution.
I would appreciate hints about what is wrong with this solution.
Thanks!

 

[Valkarie]

\begin{align*} &\text{Let a tapered beam with an elliptic cross section of length } L \text{ be fixed at one end and a force F is applied to its tip at the free end.} \\& \text{The area moment of inertia } I \text{ is a linear function of the beam length }L.\\&\text{Assuming that the deflection }y(x) \text{ is non-linear, we apply the relation between second derivative of deflection and moment:}\\&\frac{d^2y}{dx^2}= \frac{M(x)}{EI}\\&\text{Integrating twice, the general solution for the deflection is given by: }\\&y(x)= C_1 x + C_2 + \frac{M_0x^2}{2EI}+ \frac{1}{EI} \int_0^x \int_0^u M(t) dt du\end{align*}To find the constants of integration, we use the boundary conditions: $y(0)=0$ and $y_x(L)=0$. Then,\begin{align*}&C_1=0 \text{ and } C_2=0 \\&\implies y(x)= \frac{M_0x^2}{2EI}+ \frac{1}{EI} \int_0^x \int_0^u M(t) dt du\end{align*}To find the slope at the free end, we differentiate the expression twice and set $x=L$.\begin{align*}y_x(L)=\frac{M_0L}{EI}+ \frac{1}{EI} \int_0^L \int_0^u M(t) dt du\end{align*}Now, let $a$ and $b$ be the major and minor semi-axes of the elliptic cross-section respectively. Then,\begin{align*}&\text{From the known expression for the slope of a tapered beam with circular cross-section under the same boundary conditions, we get }\\&y