differential equation

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

The objective of an ordinary differential equation is to eliminate the derivative by integration and obtain an equation:
(x^2+2xy+y^2)dy=a^2dx
(2xy+y^2)dy =(a^2)dx/(x^2)
(2y+y^2)dy =(a^2)dx/(x^3)
y^2+(y^3)/3+c1 =a^2(-1/2x^2)+c2
Subtract c1 from both sides
y^2+(y^3)/3=a^2(-1/2x^2)-c1+c2
Now solve for y and consolidate -c1 and c2 into a single constant c, like this: -c1+c2=c
You may want to check my steps. Good luck.
 
The objective of an ordinary differential equation is to eliminate the derivative by integration and obtain an equation:
(x^2+2xy+y^2)dy=a^2dx
(2xy+y^2)dy =(a^2)dx/(x^2) <<<<< This step is not correct
(2y+y^2)dy =(a^2)dx/(x^3)
y^2+(y^3)/3+c1 =a^2(-1/2x^2)+c2
Subtract c1 from both sides
y^2+(y^3)/3=a^2(-1/2x^2)-c1+c2
Now solve for y and consolidate -c1 and c2 into a single constant c, like this: -c1+c2=c
You may want to check my steps. Good luck.

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

One of the ways I can think of solving this is

u(x) = y(x) + x

u' = y' + 1

Then DE becomes:

u^2 * (u'-1) = a^2

u' = 1 + a^2/u^2

du/(1 + a^2/u^2) = dx

Now the variables are separated - integrate away....
 
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