Use the transformation

$$y(t)=\frac{1}{c x(t)} \frac{d x(t)}{d t}$$

where $c$ is a constant, to map the Ricatti equation

$$\frac{d y}{d t}+c y^{2}+a(t) y+b(t)=0, \quad t>0$$

to a linear equation.

Using the above result, as well as the change of variables $\tau=\ln t$, solve the boundary value problem

$$\begin{gathered} \frac{d y}{d t}+y^{2}+\frac{y}{t}-\frac{\lambda^{2}}{t^{2}}=0, \quad t>0 \\ y(1)=2 \lambda \end{gathered}$$

where $\lambda$ is a positive constant. What is the value of $t>0$ for which the solution is singular?