The Kähler condition requires $D^{2}u$ to be positive definite, and $d_{1}\frac{\partial u}{\partial x_{1}}+d_{2}\frac{\partial u}{\partial x_{2}}>0$. These are equivalent to

$$v^{\prime\prime}>0,\quad(n+2)vv^{\prime\prime}-2v^{\prime 2}>0,\quad\frac{n+2}{n}v+(1-t)v^{\prime}>0.$$

The first two inequalities imply $v>0$. These constraints are all quite natural in view of the ODE.