TOPIC: DIFFERENTIAL EQUATION

Please answer the following questions without using the undetermined coefficient method of differential equations.

QUESTION 1:
Use the substitution v = x + y + 3 to solve the following initial value problem:
dy/dx = (x + y + 3)².

QUESTION 2:
Solve the following homogeneous differential equation:
(x² + y²) dx + 2xy dy = 0.

QUESTION 3:
Show that the differential equation:
y² dx + (2xy + cos y) dy = 0
is exact and find its solution.

QUESTION 4:
Solve the following differential equation:
dy/dx = 2y / x - (x²y²).

QUESTION 5:
Use the method of undetermined coefficients to solve the differential equation:
d²y/dt² + 9y = 2cos(3t).