Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) §6.7 Hyperbolic Functions, #57 · Calculus · Differentiation rules

A solitary water wave has a shape satisfying the … equation

A solitary water wave has a shape satisfying the $K d V$ equation $y^{''} = y^{'} - 6 y y^{'}$ .
(a) Integrate once to find $y^{''}$ . Multiply the answer by $y^{'}$ .
(b) Integrate again to find $y^{'}$ (all constants of integration are zero).
(c) Show that $y = \frac{1}{2} \sech^{2} (x /2)$ gives the shape of the "soliton."

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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §6.7 Hyperbolic Functions, #57

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