![]() The differential equation have irreducible differential Form.The general first order ODE in this case were of the form: dy dx = f (x, y) = 1 P(x, y), where P(x, y) is a polynomial in x and y such that P(x, y) = a1,0x a0,1y a1,1xy a0,2x2 aw1,w2xw1yw2, where ai,j is limited for each i, j 0 and w1, w2 are unlimited with 1 wp /2 z ? Microhal(0) and z is an infinitesimal, for p The differential coefficients are infinitesimals or unlimited and the first order differential equation have the form: dy ω1,ω2 i j xdx = f (x, y) = αx βy ∑ i,j=0 i j≥2 ai,jx y, where ai,j is limited for each i, j ≥ 0 and ω₁, ω₂ are unlimited with 1, 1 ∈ (ζ − Microhal(0))ᶜ. Furthermore, Nonstandard analysis tools are successfully applied to find a Nonstandard analytic solution for the first order differential equation near singularity in the following cases: The differential equation with one of the coefficients is unlimited. ![]() Additionally, we wrote the Legendre polynomial on the form of Mehler-Dirichlet Integral formula to find its solutions near the singularity. Then the solutions of second order ordinary differential equation (Legendre Equation) are found around the singularity in the monad of zero by using power series method with suitable transformations for singular points. We analyzed and proved the Existence and Uniqueness Theorems for first order ordinary differential equations in the subset of the monad of the initial standard point. In this thesis, we presented some Nonstandard concepts to study the analyticity near the singularity.
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