Exact solutions of a third-order ODE from thin film flow using λ-symmetry method

• We obtained exact implicit solutions of the third order ordinary differential equation y'''=y−ny'''=y−n (when n=(5/2) and n=(5/4)).
• The contact angle ∅=90° when n=(5/2).
• When n=(5/4), the contact angle ∅ varies with the values of the initial conditions.
• The value of the contact angle ϕ (when n=(5/4)) depends on the position of the contact line x⁎.

We obtain exact implicit solutions of the third order ordinary differential equation y'''=y−ny'''=y−n for the first time. These exact solutions are obtained using λ-symmetry reduction method, when n=(5/2) and n  =(5/4). The obtained solutions are valid for the general initial conditions y(α)=γ,y′(α)=δ,y″(α)=β. It is shown that the contact angle ∅=90° when n=(5/2), and it varies with the initial conditions when n=(5/4).