Using the ideas of supersymmetric quantum mechanics, we exactly solve a continuous family of anharmonicpotentials, which are the supersymmetric partners of the linear harmonic oscillators. The family includes a series ofpotentials in which the excited-state energy is the same as that of the harmonic oscillators, but the ground-state energycan be any value lower than the excited states. The shape of the potential is variable, which includes the double-welland triple-well potentials. All the potentials obtained in this paper are free of singularities, and the supersymmetry ofthe solutions is unbroken.