Differential Equation Theorems And Explanations
As I have discussed previously, there are many methods and related theorems to solve a differential equation. So now I will discuss all theorems and applications related to it.It may be longer but beneficial for solving problems of differential equations. Theorem_1 : The differential equation M(x,y) dx + N(x,y)dy= 0 is exact iff ∂M/∂y= ∂N/∂x Proof : If the given equation is exact , we have d(u(x,y))= M(x,y) dx + N(x,y) dy. ... (1) and we know that d(u(x,y)) = ∂u/∂x dx + ∂u/∂y dy .......(2) Consequently by comparison of (1) and(2) ∂u/ ∂x= M(x,y) , ∂u/ ∂y = N(x,y) .........(3) Moreover, ∂M/∂y=∂²u/ ∂y∂x and ∂N/∂x=∂²u/ ∂x∂y and because of ∂²u/∂y∂x =∂²u/∂x∂y, we get ∂M/∂y=∂N/∂x ..........(4) Convesly , let ∂M/∂y= ∂N/∂x and we shall prove that M(x,y) dx + N(x,y) dy is an exact differential equation. Let V (x,y) = ∫ M dx, Here we have integrate