MTH166 NOTESUNIT 1Exact Differential Equation1. A Differential Equation of the form M(x,y)dx+N(x,y)dy=0 is an exact differential method if ???? =???? . 2. A Differential Equation of the form M(x,y)dx+N(x,y)dy=0 is a non-exact differential method if ???? ≠???? .3. A Differential Equation of the form yf(x,y)dx+xg(x,y)dy=0 is a non-exact differential method if ???? ≠???? . The equation y = px + f (p) is known as Clairaut’s equation.UNIT-2?0 ? ?? + ?1 ? ??-1 + ⋯ + ??-1 ? ?′ + ?? ? ? = ? ?A linear ordinary differential equation of order n, is written as:Theorem: if the function ?0 ? ?1 ? ?? ? and ? ? are continuous over Iand ?0 ? ≠ on I, then there exist a unique solution to intial valueproblem?0 ? ?? + ?1 ? ??-1 + ⋯ + ??-1 ? ?′ + ?? ? ? = ? ?? ?0 = 1 ?′ ?0 = ??-1 ?0 = ?Where ?0 and 1 ? are n known constants. Different Case and MethodTo find CF. CaseSolutionIf roots of AE are real and distinct thenIf two roots are real and equal thenIf three roots are real and equal thenIf roots are complex and occur inconjugate pairIf Complex roots are equal LINEAR DEPENDENT AND LINEAR INDEPENDENTUNIT-3CASES TO CALCULATE PI ?0???? + ?1??-1 ??-1 + ⋯ + ??-1??′ + ??? = ? ?Euler-Cauchy Equation QUESTIONS
