Answer:
Explanation:
To solve the homogeneous differential equation:
dy/dx = (y^2 + x^2)/x^2
We first need to rewrite it in a homogeneous form by dividing both sides by y^2:
dy/dx = (1 + x^2/y^2)/(x^2/y^2)
Let u = y/x, then we can substitute y = ux and dy/dx = u + x(du/dx) into the differential equation to get:
u + x(du/dx) = (1 + u^2)/x^2
Rearranging and separating variables:
x(du/dx) = (1 + u^2 - u)/x
dx/x = (1 - u + u^2)/x * du
Integrating both sides:
ln|x| = ∫(1 - u + u^2)/u du + C
ln|x| = u - 0.5ln|u^2 - u + 1| + C
Substituting back u = y/x:
ln|x| = y/x - 0.5ln|(y/x)^2 - y/x + 1| + C
Simplifying:
ln|x| = y/x - 0.5ln|y^2 - xy + x^2| + C
Exponentiating both sides:
|x| * e^(y/x) = Ce^(-0.5ln|y^2 - xy + x^2|)
|x| * e^(y/x) = C|y^2 - xy + x^2|^-0.5
Squaring both sides and simplifying:
(x^2 + y^2)e^(2y/x) = D
where D = C^2. This is the general solution to the homogeneous differential equation.