For the Cauchy problem for the nonlinear infiltration equation $$\left\{\begin{array}{@{}l@{\qquad}l} u_{t}=\frac{1}{m}(u^{m})_{xx},&x\in{\mathbb{R}}, t>0,m\geq{}1,\\[3pt] u|_{t=0}=u_{0}(x),&x\in{\mathbb{R}}, \end{array} \right.$$ we use its linear solution $u(x,t,1)$ to approach the nonlinear solution $u(x,t,m)$, and obtain the explicit estimate: $$\int_{0}^{T}\int_{\mathbb{R}}|u(x,t,m)-u(x,t,1)|^{2}\,dx\,dt{} \leq{}(C^{\ast}(m-1))^{2},$$ where $C^{\ast}=O(T^{\gamma})$ and $\gamma=\frac{1+m-\alpha}{2(1+m)}$ for any $0<\alpha<1$.