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<== Differential Equations
Number Question Answer
See the work

1.

 

 

Which of the following functions are solutions of the differential equation y''-6y'+8y=0?

A. y(x)=e^(2x) 
B. y(x)=e^(-x) 
C. y(x)=e^(x) 
D. y(x)=e^(4x) 
E. y(x)=2x 
F. y(x)=4x 
G. y(x)=0
A. y(x)=e^(2x)
D. y(x)=e^(4x)
G. y(x)=0
The work

Point Cost: 3

2.

 

 

Match each of the following differential equations with a solution from the list below.

1. 2x^(2)y''+3xy'=y
2. y''-5y'+6y=0
3. y''+y=0
4. y''+5y'+6y=0
A. y=e^(2x)
B. y=e^(-3x)
C. y=(1/x)
D. y=cos(x)

1. C
2. A
3. D
4. B

The work

Point Cost: 3
3. Match each differential equation to a function which is a solution.
FUNCTIONS
A. y=3x+x^2,
B. y=e^(-5x),
C. y=sin(x),
D. y=x^(1/2),
E. y=5exp(2x),
DIFFERENTIAL EQUATIONS

1. y''+12y'+35y=0
2. y'=2y
3. 2x^2y''+3xy'=y
4. xy'-y=x^2

1. B
2. E
3. D
4. A

The work

Point Cost: 3
4. Match the following differential equations with their solutions.
The symbols A, B, C in the solutions stand for arbitrary constants.
You must get all of the answers correct to receive credit.

1. (d^2y)/(dx^2)+16=0
2. dy/dx=(-2xy)/(x^2-4y^2)
3. (d^2y)/(dx^2)+6(dy/dx)+9y=0
4. dy/dx=8xy
5. dy/dx+9x^2y=9x^2

A. 3yx^2-4y^3=C
B. y=Ae^(4x^2)
C. y=Ce^(-3x^3)+1
D. y=Acos(4x)+Bsin(4x)
E. y=Ae^(-3x)+Bxe^(-3x)

1. D
2. A
3. E
4. B
5. C

The work
5. Find the value of k for which the constant function x(t)=k is a solution of the differential equation

7t^2(dx/dy)+2x-7=0.

7/2
The work

Point Cost: 3
6. For what values of r does the function y=2e^(rx) satisfy the differential equation
y''-13y'+36y=0? The smaller one is .
The larger one (possibly the same) is .


r= 4 smalller
r= 9 larger
The work

Point Cost: 3
7. Find k such that x(t)=18^(t) is a solution of the differential equation dx/dt=kx.
k= .
ln(18)
The work

Point Cost: 3
8. For what positive values of k does the function y=sin(kt) satisfy the differential equation y''+361y=0?


For what negative values of k does the function y=cos(kt) satisfy the differential equation y''+361y=0?
k= 19
k= -19
The work

Point Cost: 3
9. It is easy to check that for any value of c, the function
y=ce^(-2x)+e^(-x)
is solution of equation
y'+2y=e^(-x)
Find the value of c for which the solution satisfies the initial condition y(2)=6.
c=


C= (6-e^(-2))/(e^-4)
The work

Point Cost: 3
10. It is easy to check that for any value of c, the function
y=x^2+c/(x^2)
is solution of equation
xy'+2y=4x^2, (x>0)
Find the value of c for which the solution satisfies the initial condition y(8)=9.
c=
C=-3520
The work

Point Cost: 3
11. The solution of a certain differential equation is of the form
y(t)=aexp(7t)+bexp(12t)
where a and b are constants.
The solution has initial conditions y(0)=3 and y'(0)=4
Find the solution by using the initial conditions to get linear equations for a and b.

y(t)=
(32/5)*e^(7t)+(-17/5)e^(12t)
The work

Point Cost: 3