ABSTRACT
Here, L is an arbitrary linear differential operator (of any order) with respect to the spatial variables x1, . . . ,xn. 1◦. Solution:
u = ϕ(t) + b exp [∫
f (t, aϕ-bψ) dt ] θ(x, t), w = ψ(t) +a exp
[∫ f (t, aϕ-bψ) dt
] θ(x, t),
where ϕ =ϕ(t) and ψ =ψ(t) are determined by the system of ordinary differential equations ϕ′t = ϕf (t, aϕ – bψ) + g(t, aϕ – bψ), ψ′t = ψf (t, aϕ – bψ) + h(t, aϕ – bψ),
and the function θ = θ(x1, . . . ,xn, t) satisfies linear equation ∂θ
∂t = L[θ].
