Đáp án+Giải thích các bước giải:

$1)$

$(x^2 + y)dx = (2y - x)dy$

$\Leftrightarrow x^2dx + ydx = 2ydy - xdy$

$\Leftrightarrow x^2dx + ydx + xdy - 2ydy = 0$

$\Leftrightarrow x^2dx + (ydx + xdy) - 2ydy = 0$

$\Leftrightarrow d\left(\dfrac{x^3}{3}\right) + d(xy) - d(y^2) = 0$

$\Leftrightarrow d\left(\dfrac{x^3}{3} + xy - y^2\right) = 0$

$\Leftrightarrow \dfrac{x^3}{3} + xy - y^2 = C$

$2)$

$(e^y + 4)dx + (xe^y + 5)dy = 0$

$\Leftrightarrow e^ydx + 4dx + xe^ydy + 5dy = 0$

$\Leftrightarrow (e^ydx + xe^ydy) + 4dx + 5dy = 0$

$\Leftrightarrow d(xe^y) + d(4x) + d(5y) = 0$

$\Leftrightarrow d(xe^y + 4x + 5y) = 0$

$\Leftrightarrow xe^y + 4x + 5y = C$

$3)$

Đề lỗi -> Sửa đề thêm "y"

$(1 + y\cos(xy))dx = -x\cos(xy)dy$

$\Leftrightarrow dx + y\cos(xy)dx + x\cos(xy)dy = 0$

$\Leftrightarrow dx + \cos(xy)(ydx + xdy) = 0$

$\Leftrightarrow dx + \cos(xy)d(xy) = 0$

$\Leftrightarrow d(x) + d(\sin(xy)) = 0$

$\Leftrightarrow d(x + \sin(xy)) = 0$

$\Leftrightarrow x + \sin(xy) = C$

$y(1) = 2\pi$

$\Rightarrow 1 + \sin(1 \cdot 2\pi) = C$

$\Leftrightarrow 1 + 0 = C$

$\Leftrightarrow C = 1$

$\Rightarrow x + \sin(xy) = 1$