Prudenciano

Consider the second order IVP

y''+3y'+2y=g(t), y(0)=0, y'(0)=0

i) SOLVE y(t) as y(t) = h(t)*g(t) and write down h(t)

ii) For g(t) given by g(t) =

0, 0<=t<3,

1, 3<=t

CALCULATE h(t)*g(t)

Use formulas in the Laplace Transform table to compute

L^(-1) { [(s+1) / (s^(2))] * (e^(-s)) }

Use Laplace Transformation to Solve the IVP

y''+4y'+4y= g(t), y(0)= 0, y'(0)= 1

g(t) =

0 , 0 < t < 1

t , 1 >= t < 3

0 , 3 =< t

Solve the IVP

y''+4y'+4y=g(t), y(0)=0, y'(0)=1

g(t)=

Function 1: 0, Interval: 0 < t < 1

Function 2: t, Interval: 1 >= t < 3

Function 3: 0, Interval: 3 <= t

Use formulas in the Laplace Transform table to compute

I) L {(t^(2)-2t+1)H(t-3)}

II) L {f(t)} for f(t) given by

Function 1: 0, Interval: 0 <= t < pi

Function 2: cos(t-(3pi/2)), Interval: pi <= t

Laplace Table:

http://tutorial.math.lamar.edu/pdf/Laplace_Table.p...

Given y″−8y′+16y= (e^(4x)) / (1+x^2)

(1) Find a fundamental set for y″−8y′+16y=0

(2) For our particular problem we have W(x)=_____

u1=∫ (−y2(x)f(x)) / W(x) dx =∫ _____

dx= ____

u2=∫ (y1(x)f(x)) / W(x) dx=∫ _____

dx= _____

Combining these Yp=_____

(3) Finally, the general solution is y=yc+yp where yc=ay1+by2 where a and b are arbitrary constants. Use the general solution to find the unique solution of the IVP with initial conditions y(0)=5 and y′(0)=−4.

y= ______

We consider the initial value problem 9x^(2)y″+9xy′+y=0, y(1)=3, y′(1)=2

By looking for solutions in the form y=x^r in an Euler-Cauchy problem Ax^(2)y″+Bxy′+Cy=0, we obtain a auxiliary equation Ar2+(B−A)r+C=0 which is the analog of the auxiliary equation in the constant coefficient case.

(1) For this problem find the auxiliary equation: _______ =0

(2) Find the roots of the auxiliary equation:_______ (enter your results as a comma separated list )

(3) Find a fundamental set of solutions y1,y2: _______(enter your results as a comma separated list )

(4) Recall that the complementary solution (i.e., the general solution) is yc=c1y1+c2y2. Find the unique solution satisfying y(1)=3, y′(1)=2

y= _________

We consider the non-homogeneous problem y″−y′=30cos(3x)

4) Apply the method of undetermined coefficients to find yp=

5) Given the initial conditions

y(0)=0 and y′(0)=−2 find the unique solution to the IVP

Find the general solution of

y″−8y′+25y=0

Please show the steps as well.

In a countercurrent heat exchanger, methanol vapor flowing at 5.50

SCMM (standard cubic meters per minute) is heated from 65 C to

260 C. What is the enthalpy change?

Suppose you extract heat from saturated steam at 300 Celsius entering a

heat exchanger in which the steam condenses and leaves the heat exchanger

as saturated liquid water at 90 Celsius.

(a) For a total enthalpy change of

DH = m˙ Dh = -44.700kJ/s what

is the required mass flow rate of the steam?

(b) What is the volumetric flow rates, m3/min in and out of the exchanger

1) Find the value of the constant C and the exponent r so that y=Cx^r is the solution of this initial value problem.

2) Determine the largest interval of the form a<x<b on which the existence and uniqueness theorem for first order linear differential equations guarantees the existence of a unique solution.

a<x<b is

3) It can happen that the interval predicted by the Fundamental Theorem is smaller than the actual interval of existence. What is the actual interval of existence in the form a<x<b for the solution (from part 1)?

a<x<b is

Please show me the steps on how to solve #1.

The general solution of the equation

y″+9y=0 is y=c1cos(3x)+c2sin(3x).

Find values of c1 and c2 so that y(0)=−1 and y′(0)=12