= Solution
For a <Gaussian function>, the required <derivatives> are
$$
v'=-2cyv,\qquad v''=(-2c+4c^2y^2)v.
$$
Substitute these into the reduced <ordinary differential equation> and compare the constant and quadratic coefficients:
$$
-\frac12+cy^2=\lambda(-2c+4c^2y^2).
$$
The constant terms give $c=1/(4\lambda)$, and this value also satisfies the quadratic terms. Hence
$$
\boxed{c=\frac1{4\lambda},\qquad
u(x,t)=\frac1{\sqrt{t+a}}
\exp\left[-\frac{(x+b)^2}{4\lambda(t+a)}\right],\qquad t>-a.}
$$
A constant multiple is also a solution by <linearity>.
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