# Find a particular solution of the differential equation $$(x-y)(dx+dy) = dx - dy$$,given that $$y=\;-1$$,when $$x=\;0$$

$(Hint:put\; x-y=\;t)$

Toolbox:
• A linear differential equation of the form $\large\frac{dy}{dx}$$=F(x,y) where F(x.y) is of the form g(x).h(y) where g(x) is a function of x and h(y) is a function of y,then the equation is variable separable type.Such equations can be solved by separating the variables. Step 1: Given :(x-y)(dx+dy)=dx-dy On expanding and rearranging we get, x(dx-dy)-y(dx-dy)=dx-dy (x-y+1)dy=(1-x+y)dx \large\frac{dy}{dx}=\frac{(1-x+y)}{(x-y+1)} Step 2: Let (x-y)=t then 1-\large\frac{dy}{dx}=\frac{dt}{dx} or \large\frac{dy}{dx}=$$1-\large\frac{dt}{dx}$
$1-\large\frac{dt}{dx}=\frac{(1-t)}{(1+t)}$
$-\large\frac{dt}{dx}=\frac{(1-t)}{(1+t)}$$-1 -\large\frac{dt}{dx}=\frac{-2t}{(1+t)} \large\frac{dt}{dx}=\frac{2t}{(1+t)} Separating the variables \large\frac{(1+t)dt}{2t}$$=dx$