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1)

Just as there are simultaneous algebraic equations (where a pair of numbers have to satisfy a pair of

equations) there are systems of differential equations, (where a pair of functions have to satisfy a pair of differential equations).

Indicate which pairs of functions satisfy this system. It will take some time to make all of the calculations.

y′_1=y_1−2y_2            y′_2=3y_1−4y_2

A. y_1=cos(x)     y_2=−sin(x) 

B. y_1=sin(x)+cos(x)      y_2=cos(x)−sin(x)

C. y_1=e^(4x)    y_2=e^(4x)


D. y_1=e^(−x)      y_2=e^(−x)


E. y_1=2e^(−2x)     y_2=3e^(−2x)

F. y+1=sin(x)      y_2=cos(x)

G. y_1=e^x     y_2=e^x

2)

Write the given second order equation as its equivalent system of first order equations.

u″+7u′+3u=0
Use v to represent the "velocity function", i.e. v=u′(t).
Use v and u for the two functions, rather than u(t) and v(t). (The latter confuses webwork. Functions like sin⁡(t) are ok.) 

u′=?

v′=?
Now write the system using matrices: 

(d/dt) [ u _ v] = [????] [ u _ v]


3) 

Write the given second order equation as its equivalent system of first order equations.

u″+2u′−5.5u=−1.5sin(3t),   u(1)=1.5,     u′(1)=3
Use v to represent the "velocity function", i.e. v=u′(t)
Use v and u for the two functions, rather than u(t) and v(t). (The latter confuses webwork. Functions like sin⁡(t) are ok.) 

u′=?

v' = ?

Now write the system using matrices: 

(d/dt) [ u _ v] = [????] [ u _ v] + [???]

and the initial value for the vector valued function is: 

[u1 _ v1] = [???]


4)

Write the given second order equation as its equivalent system of first order equations.

t^2u″ + 7.5tu′ + (t2−1)u = −2sin(3t)
Use v to represent the "velocity function", i.e. v=u′(t)
Use vv and u for the two functions, rather than u(t) and v(t). (The latter confuses webwork. Functions like sin⁡(t) are ok.) 
u′=  
v′=

Now write the system using matrices: 

(d/dt) [ u _ v] = [????] [ u _ v] + [???]


5)

Consider two interconnected tanks as shown in the figure above. Tank 1 initial contains 60 L (liters) of water and 125 g of salt, while tank 2 initially contains 60 L of water and 280 g of salt. Water containing 20 g/L of salt is poured into tank1 at a rate of 1.5 L/min while the mixture flowing into tank 2 contains a salt concentration of 30 g/L of salt and is flowing at the rate of 1 L/min. The two connecting tubes have a flow rate of 5 L/min from tank 1 to tank 2; and of 3.5 L/min from tank 2 back to tank 1. Tank 2 is drained at the rate of 2.5 L/min.

You may assume that the solutions in each tank are thoroughly mixed so that the concentration of the mixture leaving any tank along any of the tubes has the same concentration of salt as the tank as a whole. (This is not completely realistic, but as in real physics, we are going to work with the approximate, rather than exact description. The 'real' equations of physics are often too complicated to even write down precisely, much less solve.)

How does the water in each tank change over time?

Let p(t)p(t) and q(t)q(t) be the amount of salt in g at time t in tanks 1 and 2 respectively. Write differential equations for p and q. (As usual, use the symbols p and q rather than p(t) and q(t).) 

p′=

q′=

Give the initial values: 

 [p(0) _ q(0)] = [????]


6)

Consider the system of differential equations

dx/dt=−3y      (dy/dt)=−3x..

Convert this system to a second order differential equation in y by differentiating the second equation with respect to t and substituting for x from the first equation.

Solve the equation you obtained for y as a function of t; hence find xx as a function of t. If we also require x(0)=4 and y(0)=5, what are x and y?

x(t)=?

y(t)=?

 7)

Let w be the number of worms (in millions) and r the number of robins (in thousands) living on an island. Suppose ww and r satisfy the following differential equations, which correspond to the slope field shown below.

dw/dt= w−wr,     dr/dt= −r+wr.

Assume w=3and r=2 when t=0
Does the number of worms increase, decrease, or stay the same at first?  ? increases decreases stays the same  
Does the number of robins increase, decrease, or stay the same at first?  ? increases decreases stays the same 

What happens in the long run? 
 ? w and r both go to zero w goes to zero and r increases to infinity w increases to infinity and r goes to zero w and r both go to stable long-term values w and r oscillate 


8)

Consider a conflict between two armies of x and y soldiers, respectively. During World War I, F. W. Lanchester assumed that if both armies are fighting a conventional battle within sight of one another, the rate at which soldiers in one army are put out of action (killed or wounded) is proportional to the amount of fire the other army can concentrate on them, which is in turn proportional to the number of soldiers in the opposing army. Thus Lanchester assumed that if there are no reinforcements and t represents time since the start of the battle, then x and yyobey the differential equations

dx/dt=−ay,    dy/ dt=−bx,

where a and b are positive constants.

Suppose that a=0.04 and b=0.01, and that the armies start with x(0)=45 and y(0)=19 thousand soldiers. (Use units of thousands of soldiers for both xx and yy.)

(a) Rewrite the system of equations as an equation for yy as a function of x:
dy/dx=

?


(b) Solve the differential equation you obtained in (a) to show that the equation of the phase trajectory is

0.04y^2−0.01x^2=C,

for some constant C. This equation is called Lanchester's square law. Given the initial conditions x(0)=45  and y(0)=19, what is C?

C=

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Sol: (1)
(A) No
(B) No
(C) No
(D) Yes
(E) No
(F) No
(G) No Sol: (2)
Let v u t u v
Then u v
Substitute for both u and u to get :
v 7v 3u 0
Solve for v :
v 3u 7v
Now
u v
v 3u 7v
d u 0 1 u dt v 3 7 v...

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