Exact Solutions of the Nonlinear Equation ${u}_{tt}=a(t){uu}_{xx}+b(t){u}_x^2+c(t)u$

We determine ansätzes that reduce the equation ${u}_{tt}=a(t){uu}_{xx}+b(t){u}_x^2+c(t)u$ to a system of two ordinary differential equations. It is also shown that the problem of construction of exact solutions of this equation of the form u = μ₁(t)x² + μ₂(t)x^α, α ∈ R, reduces to the integration of a system of linear equations ${\mu}_1^{{\prime\prime} }={\Phi}_1(t){\mu}_1,{\mu}_2^{{\prime\prime} }={\Phi}_2(t){\mu}_2,$ where Φ₁(t) and Φ₂(t) are arbitrary given functions.