In a famous textbook example of a Bayesian-Nash equilibrium, there is a first-price auction with two independent players. Each player $i$ values the item as $v_i$, which is distributed uniformly in $[0,1]$. It is assumed that the strategy of each player $i$ is to bid a fraction of his value, i.e:
$$ b_i(v_i) := a_i \cdot v_i$$
for some constant $a_i$. Then, we can conclude that, for every $a_1$, the best response of player 2 is to pick $a_2=1/2$. Hence, the only Bayes-Nash equilibrium of this form is $a_1=1/2, a_2=1/2$.
Now, I tried to slightly generalize the example by assuming the values of the players are distributed uniformly in $[c,d]$, for some constants $d>c>0$. But I did not find an equilibrium. Suppose player 1 bids $ a_i \cdot v_i$ and player 2 bids $ $a_2 \cdot v_2$. Then the expected utility of player 2 is:
$$ProbOfWinning(a_2)\cdot GainWhenWinning(a_2)$$
$$=Pr[a_1 v_1 < a_2 v_2]\cdot (1-a_2)v_2$$
$$=\frac{(a_2 v_2 /a_1)-c}{d-c}\cdot (1-a_2)v_2$$
Taking the derivative of this expression w.r.t $a_2$ gives that the best response of player 2 is:
$$ a_2 = {1\over 2} + {a_1 c\over 2 v_2} $$
so the best $a_2$ depends on $v_2$. This means that the function from value to action is not linear as I initially assumed.
Do I have any mistake in the calculation? What is a Bayes-Nash equilibrium in this auction?