Cram University

Diff Eq 1.4

<== Differential Equations    Population Growth/Decay Explanation
See the work
1. Solve the separable differential equation
dy/dx=-6y,
and find the particular solution satisfying the initial condition
y(0)=9

y(x)= .
9e^(-6x)
The work

Point Cost: 3
2. Solve the separable differential equation
y'=sqrt(2y(x)+18)
and find the particular solution satisfying the initial condition
y(-4)=9

y(x)= .
((x+10)^2-18)/2
The work

Point Cost: 3
3. Solve the separable differential equation
dy/dx=(-0.5)/(cos(y))
and find the particular solution satisfying the initial condition
y(0)=π/6

y(x)= .
arcsin(-.5x+sin(π/6))
The work

Point Cost: 3
4. Find a function y of x such that
3yy'=x and y(3)=10
y=
sqrt((x^2+291)/3)
The work

Point Cost: 3
5. The differential equation
dy/dx=(4x+2)/(21y^2+8y+8)
has an implicit general solution of the form F(x,y)=K.

In fact, because the differential equation is separable, we can define the solution curve implicitly by a function in the form
F(x,y)=G(x)+H(y)=K.

Find such a solution and then give the related functions requested.
F(x,y)=G(x)+H(y)=
7y^3+4y^2+8y-2x^2-2x
The work

Point Cost: 3
6. The differential equation
dy/dx=(21)/(y^(1/5)+49x^2y^(1/5))
has an implicit general solution of the form F(x,y)=K.

In fact, because the differential equation is separable, we can define the solution curve implicitly by a function in the form
F(x,y)=G(x)+H(y)=K

Find such a solution and then give the related functions requested.
F(x,y)=G(x)+H(y)=
.

3arctan(7x)-(5/6)y^(6/5)
The work

Point Cost: 3
7. Find the particular solution of the differential equation
((x^2)/(y^2-8))(dy/dx)=(1/(2y))
satisfying the initial condition y(1)=sqrt(9).
sqrt(e^(-1/x+1)+8)
The work

Point Cost: 3
8. Find the solution of the differential equation
(ln(y))^5(dy/dx)=x^5y
which satisfies the initial condition y(1)=e^2.
y= .
(e^(x^6+63)^(1/6))
The work

Point Cost: 3
9. Find ƒ(x) if y=ƒ(x) satisfies
dy/dx=28yx^6
and the y-intercept of the curve y=ƒ(x) is 3.
ƒ(x)= .
e^(4x^7+ln3)
The work

Point Cost: 3
10. Solve the separable differential equation
4x-6y(sqrt(x^2+1))(dy/dx)=0
Subject to the initial condition: y(0)=3.
y= .

sqrt((4(x^2+1)^(1/2)+23)/3)
The work

Point Cost: 3
11. Find the function y=y(x) (for x>0 ) which satisfies the separable differential equation
dy/dx=(9+15x)/(xy^2); x>0
with the initial condition y(1)=4.
y= .
(27ln(x)+45x+19)^(1/3)
The work

Point Cost: 3
12. Find the solution to the differential equation
dy/dt=0.5(y-100)
if y=65 when t=0.

y=

100-35e^(0.5t)
The work

Point Cost: 3
13. Solve the separable differential equation for u
du/dt=e^(5u+8t)
Use the following initial condition: u(0)=15.
u= .
-ln(exp(-75)+(5/8)- (5/8)*exp(8t))/5
The work

Point Cost: 3
14. Find the particular solution of the differential equation
dy/dx=(x-3)(e^(-2y))
satisfying the initial condition y(3)=ln(3).
(ln(x^2-6x+18)/2)
The work

Point Cost: 3
15. The differential equation
dy/dx=(cos(x))((y^2+10y+24)/(8y+42))
has an implicit general solution of the form F(x,y)=K.

In fact, because the differential equation is separable, we can define the solution curve implicitly by a function in the form
F(x,y)=G(x)+H(y)=K.

Find such a solution and then give the related functions requested.
F(x,y)=G(x)+H(y)= .
sin(x)-(5ln(y+4)+3ln(y+6))
The work

Point Cost: 3
16. The differential equation
dy/dx=12+24x+10y+20xy
has an implicit general solution of the form F(x,y)=K.

In fact, because the differential equation is separable, we can define the solution curve implicitly by a function in the form
F(x,y)=G(x)+H(y)=K.

Find such a solution and then give the related functions requested.
F(x,y)=G(x)+H(y)= .
ln(12+10y)/10-(x+x^2)
The work

Point Cost: 3
17. Find the particular solution of the differential equation
dy/dx+ycos(x)=3cos(x)
satisfying the initial condition y(0)=5.
3+2e^(-sin(x))
The work

Point Cost: 3
18. Find the solution to the differential equation
4(du/dt)=u^2,
subject to the initial conditions u(0)=5.
u=
-4/(t-(4/5))
The work

Point Cost: 3
19. Solve the differential equation
dx/dt=(5x*lnx)/t
Assume x, t>0, and use the initial condition x(1)=2.
x=
2^(t^5)
The work

Point Cost: 3
20. A bacteria culture starts with 780 bacteria and grows at a rate proportional to its size. After 6 hours there will be 4680 bacteria.

(a) Express the population after t hours as a function of t.
population: (function of t)

(b) What will be the population after 6 hours?

(c) How long will it take for the population to reach 2590 ?
a. 780*(e^(.2986t)
b. 4680
c. ln(2590/780)/ln(6)
The work

Point Cost: 3
21. Newton's law of cooling says that the rate of cooling of an object is proportional to the difference between the temperature of the object and that of its surroundings (provided the difference is not too large). If T=T(t) represents the temperature of a (warm) object at time t, A represents the ambient (cool) temperature, and k is a negative constant of proportionality, which equation(s) accurately characterize Newton's law?
A. dT/dt=kT(T-A)
B. dT/dt=k(T-A)
C. dT/dt=k(A-T)
D. dT/dt=kT(1-T/A)
E. All of the above
F. None of the above
B
The work
22
Newton's law of cooling states that the temperature of an object changes at a rate proportional to the difference between its temperature and that of its surroundings. Suppose that the temperature of a cup of coffee obeys Newton's law of cooling. If the coffee has a temperature of 205 degrees Fahrenheit when freshly poured, and 2.5 minutes later has cooled to 194 degrees in a room at 66 degrees, determine when the coffee reaches a temperature of 164 degrees.
The coffee will reach a temperature of 164 degrees in minutes.
10.5983412
The work

Point Cost: 3
23. A thermometer is taken from a room where the temperature is 19ºC to the outdoors, where the temperature is 1ºC. After one minute the thermometer reads 12ºC.
(a) What will the reading on the thermometer be after 2 more minutes?
,
(b) When will the thermometer read 2ºC?
minutes after it was taken to the outdoors.
a) 5.1081
b) 5.869061148
The work

Point Cost: 3
24. A curve passes through the point (0,8) and has the property that the slope of the curve at every point P is twice the y-coordinate of P. What is the equation of the curve?
y(x)=
8e^(2*x) The work
25. Water leaks from a vertical cylindrical tank through a small hole in its base at a rate proportional to the square root of the volume of water remaining. The tank initially contains 175 liters and 18 liters leak out during the first day.

A. When will the tank be half empty? t= days

B. How much water will remain in the tank after 5 days? volume = Liters
A.
B.
The work