Differential Equations
· (7 points) Find all equilibrium solutions and classify each as stable, semi-stable, or unstable. Plot the phase portrait of the solutions of this equation.
\frac{d y}{d x}=y^{3}(y+3)^{2}(y-1)
\text { s) If } y(0)=-2, \text { what is } \lim _{x \rightarrow \infty} y(x) \text { ? }
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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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