Differential Equations Answers

show that function u(x,t)=(exp^-(mu)t)sin x is a solution of of one dimensional heat equation

A block of mass 1 kg is attached to a spring of force constant k = 25/4 N/m. It is pulled x = 0.3 m from its equilibrium position and released from rest. This spring‐block apparatus is submerged in a viscous fluid medium which exerts a damping force of – 4 v (where v is the instantaneous velocity of the block). Determine of the position x(t)
of the block at time t.

Solve the following ODE using the power series method:
(x + 2)y′′ + xy' − y = 0

Identifying the differential equation and solve them
y=xy' + 1 - lny'

Newton’s Law of Cooling states that the rate at which the object's temperature decreases is directly proportional to the difference between the temperature of the object and the ambient temperature. A cup of tea (temperature = 90°C) is placed in a room whose temperature is 30°C. After five minutes, the temperature of the tea has dropped to 70°C. After how much time would the temperature of the tea be 50°C?

Solve the following ordinary differential equation:
(a) dy/dx=(y-x)/(x-4y)
(b) (2yx^2+4)dy/dx + (2y^2x-3)=0
(c) y"+3y'-10y=3x^2

Identify the following differential equations and hence solve them

1) y'=-4/x^2 -y/x +y^2
2) y= xy'+1- ln y'

The linear equation y = 4x + 90 can be used to calculate the total cost y (in pounds) for x teenagers attending if they choose the cinema. Explain why this equation holds

(2yx^2 +4)dy/dx+(2y^2x -3)=0

dy/dx=(y-x)/(x-4y)