Çözüldü First Order Linear Differential Equation-Integration Factor-Lagrange Variation of Parameters Method

From the University of Alabama - Huntsville:


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[Page 110 (8 in the pdf file), Example 5.4]

Solution by Lagrange's Variation of Parameters Method:
dy / dx + y·cot(x) = 0 ⇒ y = c1·csc(x)....(I) (trivial operations are homework for the interested students)
y' = c1'·csc(x) + c1·[-csc(x) ]·[ cot(x) ]....(II)
Substituting (II) and (I) in the full equation from left to right; c1'·csc(x) + c1·[-csc(x) ]·[ cot(x) ]
{ c1'·csc(x) + c1·[-csc(x) ]·[ cot(x) ] } + [ c1·csc(x) ]·cot(x) = x·csc(x)
c1' = x ⇒ c1 = x^2 / 2 + c2....(III)
Moving (III) to (I); y = (x^2 / 2 + c2)·csc(x) = (x^2 + c) / [ 2·sin(x) ].

Solution by Variable Change of y = u·v assuming u and v are functions of x:
Also homework for the interested students. There are tons of examples in the forum.