  (Abridged) We are conducting a survey of Galactic shell-type supernova
remnants (SNRs) known or suspected to possess non-thermal components to their
X-ray emission using new and archived observations made with such X-ray
satellites as ROSAT, ASCA, RXTE, Chandra and XMM-Newton. This research is
intended to probe the phenomenon of cosmic-ray acceleration by Galactic SNRs
and estimate the maximum energy of cosmic-ray electrons accelerated by these
sources. To illustrate this work, we examine the X-ray spectrum of the
northwestern rim of an SNR suspected to have a non-thermal component to its
X-ray emission, G156.2+5.7 (RX J04591+5147), over the energy range of
approximately 0.7-12.0 keV using observations made by the ASCA GIS and the RXTE
PCA. We compare fits made to the non-thermal component using two models, a
simple power law and SRCUT. Both models give acceptable fits: the photon index
derived from the fit made with the power law model, 2.0 (+0.2, -0.5), is
comparable to values obtained for the bright rims of other SNRs with hard X-ray
spectra. Using the SRCUT model, we derive a value of 2.42 (+0.24, -0.23) x
10^17 Hz for the cutoff frequency: based on this value and assuming a mean
magnetic field strength of 14 microGauss, we estimate the cutoff energy of
cosmic-ray electrons accelerated by G156.2+5.7 to be approximately 32 TeV. This
energy value is well short of the "knee" feature of the cosmic-ray spectrum.
