Existence and Uniqueness Theorems in an Infinitesimal Micromonad of an Initial Standard Point and Legendre Equation

Some Nonstandard Treatment of the Singularity in the Differential Equation

“…where 𝑥, 𝑦 ∈ ℝ. Existence and uniqueness theorems [7] assert that there exists a unique analytic solution 𝑦 𝑜 = 𝑓(𝑥 𝑜 ) for 𝑥 = 𝑥 0 when f(x, y) is analytic on 𝐈𝐧𝐭𝐁𝐧𝐝(|𝑥 0 | + 𝜉 , |𝑦 0 | + 𝛿) where 𝜉 and 𝛿 are positive real infinitesimals [9]. Next we discuss the analyticity of the solution of equation (2).…”

“…It is clear that (0,0) is a singular point of the differential equation (7). First, we try to show that the differential equation ( 7) has a shadow solution in the monad of its singularity (0,0), then we try to show whether (7) has a solution or not for all cases given in the hypothesis and to determine the type and behavior 10-148 of each existence solution.…”

“…First, we try to show that the differential equation ( 7) has a shadow solution in the monad of its singularity (0,0), then we try to show whether (7) has a solution or not for all cases given in the hypothesis and to determine the type and behavior 10-148 of each existence solution. Let dz dx = g(x, z) be an equivalent differential equation to (7) such that g(x, z) is meromorphic where (x, z) is infinitely close to the origin of ℂ 2 . We can write these two differential equations in the following form:…”

“…IntBnd(x 1 , x 2 , … , x n ) , we denoted to the interior of the bounded set and it is defined as [7]. Consider the following general form of first order differential equation:…”

“…Suppose that f(x, y) and ∂f ∂y are S-continuous functions defined on IntBnd(γ, κ), where γ and κ are positive real infinitesimals. Then there exists x ∈ γ − 𝑴𝒊𝒄𝒓𝒐𝒉𝒂𝒍(0) so that y = ψ(x) is a unique analytic solution to (1) [7]. For more details about nonstandard analysis see [4,5,6,9,12,13].…”