= Solution
{c}
Before contact with both walls, the <elastic rod> can grow freely, so its elastic <stretch ratio> is $\alpha=1$ and its length is its stress-free grown length,
$$
\ell(t)=L_0\gamma(t)=L_0(1+ct).
$$
The first contact occurs when $\ell(t_c)=L_1$. For the intended growth case $c>0$,
$$
\boxed{t_c=\frac{L_1/L_0-1}{c}=\frac{L_1-L_0}{cL_0}.}
$$
The condition $L_1>L_0$ ensures $t_c>0$. If $c=0$, contact never occurs; negative $c$ describes shrinkage rather than the stated continuing growth.
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