The Bateman equation is the second-order nonlinear partial differential equation
 |
(1)
|
where subscripts denote partial derivatives. For a twice continuously differentiable solution with nonzero gradient, its level curves are locally portions of straight lines.
An implicit family of solutions is
 |
(2)
|
where 
and 
are arbitrary twice continuously differentiable functions and 
is a constant (Fairlie and Leznov 1999). The implicit
function theorem applies wherever 
. For example, 
, 
, and 
give 
for 
.
Where 
,
the ratio 
satisfies
 |
(3)
|
which is an inviscid Burgers' equation after reversing the sign of one independent variable (Fairlie and Leznov 1999).
This partial differential equation is distinct from the linear systems of ordinary differential equations also called Bateman equations in radioactive-decay applications (Apelblat et al. 2021).
