Q3)A: Given H(x,y)= x²-x4 + y² as a first integral of an ODEs, find this ODES corresponding to H(x,y) and show the phase portrait by using Hartman theorem and by drawing graph of H(x,y)=c. Discuss the stability of critical points of the corresponding ODEs.

Solve the given differential equation by using an appropriate substitution. The DE is homogeneous. y-x dy dx = y + x

optımızatıon theory

optımızatıon theory level set and gradıent

QI Find the first integral + (x°) ³ + x =0

Q1: solve the system y 2 In √√x² + y2 X y = −y + In √√x² + y2 and solve the linear part.

Find all the solutions of the congruence 7x² + 15x = 4 (mod 111).

) The set {1,2,..., 22} is to be split into two disjoint non-empty sets S and T in such a way that: (i) the product (mod 23) of any two elements of S lies in S; (ii) the product (mod 23) of any two elements of T lies in S; (iii) the product (mod 23) of any element of S and any element of T lies in T. Prove that the only solution is S = {1, 2, 3, 4, 6, 8, 9, 12, 13, 16, 18}, T= {5, 7, 10, 11, 14, 15, 17, 19, 20, 21, 22}.

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