A superconducting metallic film bolometer in contact with thermal bath of temperature T0, is biased by a (i) constant current $I_{\mathrm{b} }$, or by (ii) the constant voltage $U_{\mathrm{b}}$. The absorption of the time-dependent power of a phonon beam W(t) elevates temperature of the film to $T(t)>T_{\mathrm{0}}$. The thermal contact of the bolometer with the environment is characterized by the thermal conductance G, and it depends on $[T^{n}(t)-T_{\mathrm{0}}^{n}]$ ($n=4{-}6$). For both above methods of bias, we derive ordinary nonautonomous nonlinear differential equations of the first order. These equations are numerically solved. The knowledge of W(t) allows one to calculate (i) $U_{\mathrm{b} }(t)$ or (ii) $I_{\mathrm{b}}(t)$. The inverse problem of calculation of W(t) from known (i) $U_{\mathrm{b}}(t)$ or (ii) $I_{\mathrm{b}}(t)$ is also solved. Using W(t) calculated in computer experiments, we obtained the bolometer signal, and compared it with results of real experiments performed on GaAs in which $I_{\mathrm{b}}$ is fixed, and U is measured. Comparison of calculated results with results obtained for the linearized model shows that the nonlinearity is essential for the correct description of metallic superconducting film bolometer response.