A Mathematica window showing two coupled nonlinear differential equations: 2u'[s]v'[s]/v[s]+u''[s]==0 -(1/2) v[s](u'[s])^2+v''[s]==0 Mathematic is asked to solve this system with DSolve, but cannot. A 2-parameter family of solutions: u[s_]=Sqrt[2]*ArcCos[(a*b*(Sqrt[a^2 + b^2] - a*b*s))/Sqrt[a^2*b^2*(a^2 + b^2) - 2*a^3*b^3*Sqrt[a^2 + b^2]*s + a^4*b^2*(a^2 + b^2)*s^2]]; v[s_]=Sqrt[a^(-2) - (2*b*s)/(a*Sqrt[a^2 + b^2]) + s^2]/Sqrt[2]; is checked against the equations, and shown to satisfy them.
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