演習問題1.3

1.: 次の微分方程式の一般解を求めよ．
(a): $\displaystyle{y^{\prime} = \frac{xy}{x^{2}+y^{2}}}$
(b): $\displaystyle{y^{\prime} = \frac{xy}{(x+2y)^{2}}}$
(c): $\displaystyle{y^{\prime} = \frac{x^{2}+2xy-4y^{2}}{x^{2}-8xy-4y^{2}}}$
(d): $\displaystyle{(x^{2}-y^{2}e^{\frac{x}{y}})y^{\prime} = xy }$
(e): $\displaystyle{y^{\prime} = \frac{x+2y-1}{x+2y+7}}$
(f): $\displaystyle{y^{\prime} = \frac{x-y+8}{y-3x+2}}$
: 2.] 次の初期値問題を解け．
(a): $\displaystyle{(y - \sqrt{x^{2}+y^{2}})dx - xdy = 0, y(\sqrt{3}) = 1}$
(b): $\displaystyle{(y^{3}-x^{3})dx - xy^{2}dy = 0, y(1) = 2}$
3.: 例題1.9はなぜ例題1.8の方法では解けないのか考えなさい.