Question

Differential Equations

. (i) Show that the half-range Fourier Cosine Series on the interval (0, L] of the function

f(x)=1-\frac{x}{L}

is given by

f(x)=\frac{1}{2}+\sum_{k=1}^{\infty} \frac{4}{(2 k-1)^{2} \pi^{2}} \cos \left(\frac{(2 k-1) \pi}{L} x\right) \text {. }

i) A metal bar of length L with insulated ends is initially heated. The temperature of the bar 0(x, t) is governed by the one-dimensional heat equation

\frac{\partial \theta}{\partial t}=D \frac{\partial^{2} \theta}{\partial x^{2}}

where D is a constant. It is subject to the two boundary conditions

\frac{\partial \theta}{\partial x}(x=0, t)=0 \quad \text { and } \quad \frac{\partial \theta}{\partial x}(x=L, t)=0 .

(a) Use the method of separation of variables to find the general solution in the form

\theta(x, t)=A_{0}+\sum_{n=1}^{\infty} A_{n} e^{-\beta_{n} t} \cos \left(\frac{n \pi x}{L}\right)

where the coefficients An, n = 0, 1, 2,... are unknown constants,and the form of the constants B, should be found.

(b) Find the solution where the temperature also satisfies the initial con-dition

\theta(x, t=0)=1-\frac{x}{L}

O Explain how this solution behaves as t 0o and sketch the solution of 0 against r for increasing t.

Verified

### Question 44953

Differential Equations

Consider the IVP y = x + 2y, y(0) = 1
A) (8 pts) Use Euler's method to obtain an approximation of y(0.5) usingh = 0.25 for the given IVP (Use four-decimal approximation)
B) (12 pts) Use Euler's method to obtain an approximation of y(0.5) usingh = 0.1 for the given IVP (Use four-decimal approximation)
A bacteria culture initially has 100 number of bacteria and doubles in size in 2hours. Assume that the rate of increase of the culture is proportional to the size.
Write the initial value problem for the bacteria culture and solve it
ts) How long will it take for the size to triple?
Verify that y(x) = c1 cos(6x) + c2 sin(6x) is a solution of
y" + 36y = 0
s) Either solve the boundary value problem
y^{\prime \prime}+36 y=0, y(0)=0, y\left(\frac{2 \pi}{6}\right)=1
or else show that it has no solution

### Question 44952

Differential Equations

The solution of the problem
y^{\prime}=x+2 y, y(1)=2
numerically using Euler's method for y(1.6) using h = 0.3 is
-5.99
3.5
-3.5
5.99
None of the others

### Question 44951

Differential Equations

A differential equation
y^{\prime \prime}-5 y^{\prime}+2 y=0
\text { with } y(0)=1 \text { and } y^{\prime}(0)=5 \text { is }
a second order initial value problem
a fourth order initial value problem
a third order initial value problem
None of the others
A boundary value problem

### Question 44950

Differential Equations

The solution of the initial value problem
y^{\prime \prime}+16 y=0, y(0)=0, y^{\prime}(\pi)=4
\sin 4 x
-\sin 4 x
\text { None of the others }
\cos 4 x
\cos 4 x+\sin 4 x

### Question 44949

Differential Equations

The population, P, of a town increases as the following equation:
P(t)=P_{0} e^{0.25 t}
If P(5) = 200, what is the initial population?
Select one:
Po - 59
None of the others
Po - 61
Po - 60
Po 57

### Question 44948

Differential Equations

For the following model of Newton's law of cooling:
\frac{d T}{d t}=\ln \left(\frac{1}{2}\right)(T-16), \quad T(0)=70
\text { the ambient temperature } T_{m} \text { is } 16 .
True--False

### Question 44947

Differential Equations

Consider the initial-value problem
y^{\prime}-20=y e^{2 x}, y(1)=5
Using the Euler's method we have
y_{1}=5+h\left(20+5 e^{2}\right)

### Question 44946

Differential Equations

\text { If } P(t)=2 e^{0.15 t} \text { gives the population in an environment at time } t \text {, then }
P(4)=2 e^{0.06}
O TrueO False

### Question 44945

Differential Equations

\text { The function } y=e^{-8 x} \text { is a solution of the initial value problem }
y^{\prime \prime}-64 y=0, y(0)=1, y^{\prime}(0)=8 .
True--False

### Question 44710

Differential Equations

Q2.[40pt] Draw the phase diagram of the following (system of) differential/difference equa-tions and analyze the stability of the equilibria.
\dot{x}_{t}=\frac{x_{t}}{x_{t}^{2}+1}
\dot{x}_{t}=x_{t} y_{t}
\dot{y}_{t}=-2 x_{t}-4 y_{t}+4
\dot{x}_{t}=3 x_{t}-13 y_{t}
\dot{y}_{t}=5 x_{t}+y_{t}
x_{t+1}=2-x_{t} .