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．
(a): $\displaystyle{(x^{2} + y^{2})dx + 2xy dy = 0}$
(b): $\displaystyle{(ye^{xy} + 2xy)dx + (xe^{xy} + x^{2})dy = 0}$
(c): $\displaystyle{(1 + x y^{2})dx + (x^{2}y + y)dy = 0}$
(d): $\displaystyle{(y^{2} - x^{2})dx + 2xydy = 0}$
2.: 次の初期値問題を解け．
(a): $\displaystyle{x^{2}dx + ye^{y}dy = 0, \ y(0) = 1 }$
(b): $\displaystyle{(e^{x}y + \sin{y})dx + (e^{x} + x\cos{y})dy = 0, \ y(0) = 1}$
(c): $\displaystyle{(\cos{x}\sin{x} - xy^{2})dx - y(x^{2} - 1)dy = 0, \ y(0) = 2}$