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Differential Equations

4.Divide 11 into two parts such that their product is a maximum.

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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

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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

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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

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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

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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

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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

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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)

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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

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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

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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} .

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