Math 220: First Order Scalar Semilinear Equations
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چکیده
(4) v′ r(s) = c(xr(s), yr(s), vr(s)). Note that here the subscript r denotes a parameter, not a derivative! We may equally well write xr(s) = x(r, s), yr(s) = y(r, s), and we will do so; we adopted the subscript notation to emphasize that along each integral curve r is fixed, i.e. is a constant. Which parameterization should we use? We are normally also given initial conditions along a curve Γ = Γ(r) with Γ(r) = (Γ1(r),Γ2(r)), namely
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