Differential Equations

What is the order of the differential equation \( \frac{d^2y}{dx^2} + 5\frac{dy}{dx} + 6y = 0 \)?

Which of the following is a solution to the differential equation \( \frac{dy}{dx} = 3y \)?

Find the particular solution of \( \frac{dy}{dx} = 2x \) given \( y=3 \) when \( x=0 \).

The differential equation \( \frac{dy}{dx} + y \tan x = \sin 2x \) is of which type?

Solve the differential equation \( \frac{dy}{dx} = \frac{x + y}{x} \).

If \( y = e^{2x} \) is a solution of the differential equation \( \frac{d^2y}{dx^2} - 4\frac{dy}{dx} + 4y = 0 \), what is the characteristic equation?

Find the integrating factor of the differential equation \( \frac{dy}{dx} + \frac{2}{x} y = \sin x \), \( x > 0 \).

Which of the following differential equations is exact?

Solve the differential equation \( (y + xy^2) dx + (x + x^2 y) dy = 0 \).

A tank contains 100 litres of pure water. Brine containing 2 g of salt per litre flows in at 3 litres/min and the well-mixed solution flows out at the same rate. Find the differential equation for the amount of salt \( y \) in the tank at time \( t \).