Problem 1.

1. $$y^2 = Cx + D.$$
2. $$x^2 + y^2 = C.$$
3. $$y^2 = Ce^x.$$
4. $$y = C - x.$$
5. $$\ln \lvert x \rvert = C - \tfrac{1}{x}.$$
6. $$x^2 - y^2 = C.$$
7. Two possible solutions: $$y^2 \pm 2y^2 = C.$$
8. I suspect there is a typo in the statement of this problem, because this one is pretty tough. I got it reduced to the equation $$(x-y’-y)^2 (y’)^2= y^2 \big( 1+ (y’)^2 \big)$$. We’d have to find all solutions of this equation for $$y’$$, and then solve each of those differential equations independently. ANGTFT

Problem 2.

1. $$y = 2x+C.$$
2. $$\lvert y \rvert = C \sqrt[n]{\lvert x \rvert}.$$
3. $$y^2 = \tfrac{2}{n}x^2 + C.$$
4. $$\lvert y \rvert = Cx^2.$$
5. We’ll do this one later on. Focus on the ones for which we have discussed the method $$F(x,y) = C.$$
6. $$y^2 = x + C.$$
7. Pretty tough. $$\big( \tfrac{y^2}{x^2}+1 \big) \big( \tfrac{y^2}{x^2}+2 \big)^{-1/2} = Ce^{-x}.$$
8. $$\tfrac{1}{y-x} - \ln \big( 1-\tfrac{1}{y-x} \big) = C-x.$$
9. One solution is $$y = 0.$$ The other is $$\tfrac{2}{3}x^3 + xy^2 = C.$$