Exercises 2.415.This differential equation is expressed in the variablesrandθ. Since the variablesxandyare dummy variables, this equation is solved in exactly the same way as an equation inxandy. We will look for a solution with independent variableθand dependent variabler. We seethat the differential equation is expressed in the differential formM(r, θ)dr+N(r, θ)dθ= 0,whereM(r, θ) = cosθandN(r, θ) =−rsinθ+eθ.This implies thatMθ(r, θ) =−sinθ=Nr(r, θ),and so the equation is exact.Therefore, to solve the equation we need to find a functionF(r, θ) that has cosθ dr+ (−rsinθ+eθ)dθas its total differential. IntegratingM(r, θ) withrespect torwe see thatF(r, θ) =cosθ dr=rcosθ+g(θ)⇒Fθ(r, θ) =−rsinθ+g(θ) =N(r, θ) =−rsinθ+eθ.Thus we have thatg(
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