Verify that \( y = ce^{-4x} + 0.35 \) is a solution of the ODE \( y' + 4y = 1.4 \) and determine \( c \) so that the resulting particular solution satisfies the initial condition \( y(0) = 2 \).

Solve the differential equation: \[ \frac{dy}{dx} = (x + 1)e^{-x}y^2 \]

Solve: \[ y' \sin 2\pi x = \pi y \cos 2\pi x \]

Solve \( 2xyy' = y^2 - x^2 \) by reducing it to separable form.

Solve: \[ xy' = x + y \]

Solve:
(a) \( y^3 y' + x^3 = 0 \)
(b) \( y' = e^{2x - 1}y^2 \)

Solve the differential equation: \[ xy' = y + 2x^3 \sin^2\left(\frac{y}{x}\right) \]

Solve: \[ \cos(x + y)\,dx + \bigl(3y^2 + 2y + \cos(x + y)\bigr)\,dy = 0 \]

Solve: \[ y' = Ay - By^2 \]

Find an integrating factor and solve the initial value problem: \[ (e^{x + y} + ye^y)\,dx + (xe^y - 1)\,dy = 0, \qquad y(0) = -1 \]