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We present some results of the analytical integration of the energy rate balance equation, assuming that the input energy rate is proportional to the azimuthal interplanetary electric field, <i>E<sub>y</sub></i>, and can be described by simple rectangular or triangular functions, as approximations to the frequently observed shapes of <i>E<sub>y</sub></i>, especially during the passage of magnetic clouds. The input function is also parametrized by a reconnection-transfer efficiency factor α (which is assumed to vary between 0.1 and 1). Our aim is to solve the balance equation and derive values for the decay parameter τ compatible with the observed <i>Dst</i> peak values. To facilitate the analytical integration we assume a constant value for τ through the main phase of the storm. The model is tested for two isolated and well-monitored intense storms. For these storms the analytical results are compared to those obtained by the numerical integration of the balance equation, based on the interplanetary data collected by the ISEE-3 satellite, with the τ values parametrized close to those obtained by the analytical study. From the best fit between this numerical integration and the observed <i>Dst</i> the most appropriate values of τ are then determined. Although we specifically focus on the main phase of the storms, this numerical integration has been also extended to the recovery phase by an independent adjust. The results of the best fit for the recovery phase show that the values of τ may differ drastically from those corresponding to the main phase. The values of the decay parameter for the main phase of each event, τ<i><sub>m</sub></i>, are found to be very sensitive to the adopted efficiency factor, α, decreasing as this factor increases. For the recovery phase, which is characterized by very low values of the power input, the response function becomes almost independent of the value of α and the resulting values for the decay time parameter, τ<i><sub>r</sub></i>, do not vary greatly as α varies. As a consequence, the relative values of α between the main and the recovery phase, τ<i><sub>m</sub></i>/τ<i><sub>r</sub></i>, can be greater or smaller than one as α varies from 0.1 to 1.