How would one go about solving the differential equation (x+y)(dy/dx) = x - y for y(x)?

Answer 1

You cannot integrate y alone and still have a function of y's. The previously listed answer is wrong. The D.E. you have here is an "exact" differential equation. I know this because it can be rewritten in the form: (x+y)dy-(x-y)dx=0. M(x,y)=y-x, N(x,y)=x+y. My=dM/dy=1, Nx=dN/dx=1. By dM/dy and dN/dx I mean the partial of M with respect to y or the partial of N with respect to x. Because My equals Nx, this DE is exact.

From this point, f(x,y)=Integral(x+y)dy. By dy, remember that we must use the partial differential, not the ordinary differential. When I integrate, I get f(x,y)=xy+(yy/2)+g(x). We must include this arbitrary function g(x) because we are dealing with a partial in the integral.

Now, we know that f(x,y)=xy+(yy/2)+g(x). Therefore,
g(x)=f(x,y)-xy-(yy/2). We now take the derivative of both sides with respect to x (because of the x located in g(x).). We get:
g'(x)=f'(x,y)-y. Now look at our original function! f'(x,y) is equal to x+y! Therefore, g'(x)=x+y-y. g'(x)=x. g(x)=(xx/2)+C. f(x,y)=xy+(yy/2)+xx/2+C. Let's assume that C=xy+yy/2+xx/2. Now you can find y(x).

Answer 2

it extremely is not a DE. it might probable be dy/dx=x^2/(y(a million+x^3) wherein case separate the variables to furnish dy/y=x^2 dx/(a million+x^3) combine to furnish ln|y|=(a million/3)ln|a million+x^3|+c=ln|a million+x^3|^(a million/3)+l... giving y=A(a million+x^3)^(a million/3) the place A is a relentless OR y^3=C(a million+x^3)

Answer 3

y=integral((x-y)/(x+y))dx.