Stuck trying to integrate a differential equation using substitution v=y/x

[dooogle]

Homework Statement

dy/dx=a^2/(x+y)^2

where a is a constant

need the answer in the form

x=f(y)

Homework Equations

The Attempt at a Solution

multiplying out (x+y)^2

gives dy/dx=a^2/(x^2+2xy+y^2)

setting u=y/x dy/dx can be rewritten as

dy/dx=a^2/((x^2)*(u+1)^2)

=a^2/x^2(u+1)^2

this is my question can i then use the chain rule to find du/dx where du/dx=(dy/dx)*(dx/du)

where dy/dx=a^2/((x^2)*(u+1)^2)

and dx/du=-y/(u^2)

found by differentiating x=y/u

this would give dx/du= -y*(a^2)/((x^2)*(u^2)*((u+1)^2))

thank you for your time dooogle

 

[mighty2000]

Dear dooogle,

Thank you for your question regarding the problem dy/dx=a^2/(x+y)^2. In order to find the answer in the form x=f(y), we can use the substitution method. We can let u=x+y, which means that x=u-y. Then, using the chain rule, we can rewrite the equation as:

dy/dx = dy/du * du/dx

Substituting u=x+y, we get:

dy/dx = dy/du * (1+dy/du)

Rearranging this equation, we get:

dy/du = dy/dx / (1+dy/dx)

Using the given equation dy/dx=a^2/(x+y)^2, we get:

dy/du = a^2 / (u^2)

Integrating both sides with respect to u, we get:

y = -a^2/u + C

Substituting u=x+y, we get:

y = -a^2/(x+y) + C

Rearranging this equation, we get:

x = -a^2/(y-C) - y

Therefore, the answer in the form x=f(y) is:

x = -a^2/(y-C) - y

I hope this helps. Let me know if you have any further questions.