EXPERIENCE TO EXPLAIN FIRST – ORDER EQUATION WHICH EXACT SOLUTION ARE OBTAINABLE

January 10 th 2009, at 10.00 am in Karang Malang blok B 20A.
I explain about first – order equation which exact solution are obtainable to my friend’s, she is Aida. Aida is just board with me and my classmate. In this first – order equation for which exact solution are obtainable considered certain basic types of first – order equation for with exact solution my be obtained by definite procedures. The purpose of this first – order equation for which exact solution are obtainable is to gain ability to recognize these various types and to apply the corresponding method of solution. Of the types considered here, the so called exact equation in a sense the most basic. First, I explain standard form of first –order differential equation. I given example of standard form of first order differential equation is dy/dx = f (x,y) or the differential form M (x,y) dx + N (x,y) dy = 0. An equation in one of form these may readily be written in the other form. For example the equation dy/dx = (x^2/y^2)/(x – y) is the form of (1) it may be written (x^2 + y^2) dx + (y – 1) dy = 0.Of the form (2), may be written in the form (1) as dy/dx = - (sin x + y)/(x + 3 y).
In the form (1) it is clear from the notation it self that y is regarded as the dependent variable and x as the independent one, but in the form (2) we may actually regard either variable as the dependent one and the other as the independent. However, in this text, all different differential equation of the form (2) in x and y, we shall regard y as dependent and x as independent, unless the contrary is specifically stated. For example above, she understands.
Then, I that given know definition of exact differential equation is let F be a function of two real variables such that F has continuous first partial derivatives in a domain D. the total differential dF of the function F is defined by the formula dF (x,y) = dF (x,y)/dx dx + dF (x,y)/dy dy for all (x,y) E D.
I am give example to Aida, and she can to finish true. I am happy given explanation to her. She can to finish problem mathematic with easy because she is very like mathematic. She is also university student mathematic in UNY. She is know about definition of M (x,y) dx + N (x,y) dy is called an exact differential in a domain D if there exist a function F of two variables such that is expression equal the total differential dF(x,y) for all (x,y) element D. example of the differential equation y^2 dx + 2 xy dy = 0 is an exact differential equation, since the expression y^2 dx + 2 xy dy is an differential. Indeed, it is the total differential of the function F defined for all (x,y) by F(x,y) = xy^2, since the coefficient of dx is dF(x,y)/dx = y^2 and that of dF(x,y)/dy = 2xy. On the other hand, the more simple appearing equation y dx + 2x dy = 0, obtained from y^2 dx + 2 xy = 0 by dividing through by is not y, is not exact.
I also explain about theorem of the differential equation M(x,y) dx + N(x,y) dy = 0. where M and N have continuous first partial derivative at all point (x,y) in a rectangular
1. if the differential equation M(x,y) dx + N(x,y) dy = 0 is exact in D, then dM(x,y)/dy = dN(x,y)/dx for all (x,y) element D.
2. conversely, if dM(x,y)/dy = dN(x,y)/dx
With that theorem she become very understanding manner working differential equation. Manner the solution of exact differential equation, let proceed to solve exact differential equation. If the equation M(x,y) dx + N(x,y) dy = 0 is exact in a rectangular domain D, then there exist a function F such that dF(x,y)/dx = M(x,y) and dF(x,y)/dy = N(x,y) for all (x,y) element D.
Then the equation may be written (dF(x,y)/dx) dx + (dF(x,y)/dy) dy = 0 or simply dF(x,y) = 0. the relation F(x,y) = c is obviously of this, where c is an arbitrary constant. To very know in differential equation I explain theorem again, the theorem is suppose the differential equation M(x,y) dx + N(x,y) dy = 0 satisfies the differential ability requirement of theorem above and is exact in a rectangular domain D. Then, a one parameter family of solution of this differential equation is given by F(x,y) = c, where is a function such that dF(x,y)/dx = M(x,y) and dF(x,y)/dy = N(x,y) for (x,y) element D. with theorem she more know, and I also given example to her in order to he is know. Example’s (3 x^2 + 4 xy) dx + (2 x^2 + 2 y) dy = 0, first duty is to determine whether or not the equation is exact. Here M(x,y) = 3 x^2 + 4 xy, N(x,y) = 2 x^2 + 2 y
dM(x,y)/dy = 4 x, dN(x,y)/dx = 4 x
for all real (x,y), and so the equation is exact in every rectangular domain D.
I finish explain to Aida at 11.00 am, and she is look happy with my explain.