Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2015 / Paper 1 / 13 / b / Solution

For a Gaussian function, the required derivatives areSubstitute these into the reduced ordinary differential equation and compare the constant and quadratic coefficients:The constant terms give $c=1/(4\lambda )$, and this value also satisfies the quadratic terms. HenceA constant multiple is also a solution by linearity.