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Differential Equations

x \frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}+x y=0

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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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**Use of solution provided by us for unfair practice like cheating will result in action from our end which may include permanent termination of the defaulterâ€™s account.Disclaimer:The website contains certain images which are not owned by the company/ website. Such images are used for indicative purposes only and is a third-party content. All credits go to its rightful owner including its copyright owner. It is also clarified that the use of any photograph on the website including the use of any photograph of any educational institute/ university is not intended to suggest any association, relationship, or sponsorship whatsoever between the company and the said educational institute/ university. Any such use is for representative purposes only and all intellectual property rights belong to the respective owners.

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