Differential Equations Answers

Evaluate the following functions in differential operator form.

1. D(x²+2x-3) Ans. (2x+2)
2. D²(xe^3x-e^4x) Ans. 9xe^3x+6e^3x-16e^4x

Solve the following differential equations in series.

x²d²y/dx + xdy/dx +(x²-4)y=0

reduce the following equation onto canonical form

u_x+2xyu_y=x

2x^ 3 y^ prime =y(y^ 2 +3x^ 2 )

Solve differential equation

(D^3-6D’D+11D’^2D-6D’)z=cos(2x+y)+e^(2x+y)-y

find the general solution of the homogeneous linear equations

1.(2D²-5D-3)y=0

2. (9D²-6D+1)y=0

3. (D⁴-2D²)y=0

4. (D⁵+8D³+16D)y=0

5. (D³+8)y=0

Find two members of the family of solution in problem 5 that satisfy the initial condition y(0)=0, y'(0)=0.

y''-10y'+25y=0 find general solution of the given second order differential equation.

y''-25y=0; y1=e^5x the indicated function y1(x) is a solution of the given differential eqution.Use reduction of order or formula as instructed, to find a second solution y2(x).

A metal bar with an initial temperature, 𝑇0, in the interval of 30°C ≤ 𝑇0 ≤ 35°C is dropped into a container of boiling water (100°C). The temperature of the metal bar, 𝑇 at any time, 𝑡 satisfies the following Newton’s Law of Cooling model 𝑑𝑇 𝑑𝑡 = −𝑘(𝑇 − 𝑇𝑚) where 𝑇𝑚 is the ambient temperature and 𝑘 is the constant. After 5 seconds, the temperature of the bar, 𝑇1 is in the interval of 40°C ≤ 𝑇1 ≤ 50°C. a. Find the equation that models the temperature of the metal bar, 𝑇 at any time, 𝑡 (choose a value of 𝑇0 and 𝑇1 from the given intervals, respectively). By using an appropriate analytical method, solve the derived model and explain the reason for the selection of the method. b. Compute the temperature of the metal bar after 100 seconds by using the derived model in Part 1(a) with THREE (3) different numerical methods with step size, ℎ = 10 seconds. Select the best numerical method to compute the temperature of the metal bar and justify your answer