See the attached file for questioins

1. Verify that

y=xe

2x

is a solution to the differential equation

¢¢

y-4

¢

y+4y=0

2. Given the solution

y=C

1

cosx+C

2

sinx

to the differential equation

¢¢

y+

¢

y=0

.

Find the constants C1 and C2 under the initial conditions

y

p

6

æ

è

ç

ö

ø

÷

=

1

2

and

3. Solve the differential equation

x

2

dy

dx

=y-xy

4. Find the particular solution that satisfies the initial condition.

Differential equation:

dy

dx

=4y

2

+1

()

Initial condition:

y

p

4

æ

è

ç

ö

ø

÷

=1

5. Solve the homogeneous differential equation

y

2

+yx

()

dx-x

2

dy=0

by making

the appropriate substitution.

6. The temperature of an object is 98 degrees Fahrenheit when brought into a room that is 65 degrees. Four minutes later the object cools down to 88 degrees

Fahrenheit. If the temperature of the room remains constant, how long will it take

the object to cool down to 70 degrees Fahrenheit? (Hint: Use Newton’s Law of

Cooling)

7. Solve the differential equation with the initial condition by finding an integrating factor.