Considering the cross-sectional velocity profile to be piecewise-constant in each stream of a multi-stream heat exchanger for cocurrent thermally developing flow, this study analytically solves the related conjugated Graetz problem by using an integral transform method. Further, it obtains an analytical solution in an explicit form to the fluid temperatures that vary two-dimensionally. A numerical example is provided for the case of a triple-tube heat exchanger. The numerical results demonstrate the effects of the thermal conductivity ratios of the fluids and the Peclet number ratios on the temperature distribution in the streams. In addition, in order to show the importance of considering streamwise variations in the overall heat transfer coefficients when designing laminar flow heat exchangers, the amount of exchanged heat calculated by the presented analytical solution is compared with that predicted on the basis of bulk temperatures and constant overall heat transfer coefficients.