A particle free to move along the x-axis has potential energy given by U(x)=k[1−exp(−x2)]  for −∞≤x≤+∞, where k is a positive constant of appropriate dimensions. Then

Potential energy of the particle U=k(1−e−x2)Force on particle F=−dUdx=−k[−e−x2×(−2x)]F=​ −2kxe−x2=−2kx1−x2+x42 !−......For small displacement  F=−2kx⇒F(x)∝−x   i.e. motion is simple harmonic motion.