演習問題

1. 次の微分方程式が完全微分形か調べ,完全微分形の方程式を解け．
$$\begin{array}{ll} (a) \ (x^{2} + y^{2})dx + 2xy dy = 0 & (b) ... ...}y + y)dy = 0 & (d) \ (y^{2} - x^{2})dx + 2xydy = 0 \end{array}$$
2. 次の初期値問題を解け．
$(a) \ x^{2}dx + ye^{y}dy = 0, \ y(0) = 1$
$(b) \ (e^{x}y + \sin{y})dx + (e^{x} + x\cos{y})dy = 0, \ y(0) = 1$
$(c) \ (\cos{x}\sin{x} - xy^{2})dx - y(x^{2} - 1)dy = 0, \ y(0) = 2$