Find a general solution of the differential equation

[Totalderiv]

Homework Statement

Find a general solution of the differential equation.

(x+y)y' = x-y

Homework Equations

v=x+y
y=v-x
y'=v'-1

The Attempt at a Solution

So if I plug this back into the original equation;
v(v'-1)= x-y
How do I convert v=x+y into x-y so that I have only v's and x's? Do I have to make another substitution with x-y?

 

[Dickfore]

Hint:

Make the subst:

<br /> z = \frac{y}{x}<br />
because your ode is a homogeneous equation:
<br /> y&#039; = \frac{x - y}{x + y} = \frac{\frac{x - y}{x}}{\frac{x + y}{x}} = \frac{1 - y/x}{1 + y/x}<br />

Treat z as a function of x. Exprss y in terms of x and z. Find the derivative w.r.t. x by using the chain rule. What do you get?

 

[Totalderiv]

So it would be;

y=zx
y&#039;= z+xz&#039;
z+xz&#039; =\frac {1-z}{1+z}

How do I isolate the z's and x's in order to integrate?

 

[Dickfore]

simplify:
<br /> \frac{1 - z}{1 + z} - z<br />
You will get an ODE with separated variables x and z.