Show that the DE x²(dy/dx) = y² + 2xy is homogeneous. Find its particular solution, given that y = 1 when x = 1.

1. 1 · Write what is given and what is to be found. Given: x²(dy/dx) = y² + 2xy and y(1) = 1. To show: the DE is homogeneous. To find: the particular solution.
2. 2 · Show it is homogeneous. dy/dx = (y² + 2xy)/x² = F(x, y). F(λx, λy) = λ²(y² + 2xy)/(λ²x²) = λ⁰F(x, y), so F has degree 0.
3. 3 · Write the substitution. Put y = vx, dy/dx = v + x(dv/dx). Then v + x(dv/dx) = v² + 2v.
4. 4 · Separate and integrate.
   • x(dv/dx) = v² + v, so dv/[v(v + 1)] = dx/x
   • Partial fractions: 1/[v(v + 1)] = 1/v − 1/(v + 1)
   • log|v| − log|v + 1| = log|x| + log|C|, so v/(v + 1) = Cx
5. 5 · Put back v = y/x. (y/x)/[(y + x)/x] = Cx, that is y/(x + y) = Cx.
6. 6 · Use the condition. x = 1, y = 1: 1/2 = C. So y/(x + y) = x/2, that is 2y = x² + xy.
7. 7 · Check and box the answer. y = x²/(2 − x), x ≠ 2. Check at x = 1: y = 1/1 = 1.

Scoring tip: Show the λ test in one line. Many students lose the 'show that it is homogeneous' mark by skipping it.