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**Applied Differential Equations **(MT2043)* Assignment 04*** Submission Date: 18/05/2012
**

** Instructor: Dr. Rashid Ali
**

**Q1. **A steel ball weighing 128 lb is suspended from a spring, whereupon the spring is stretched 2
ft from its natural length. The ball is started in motion with no initial velocity by displacing it 6 inch
above the equilibrium poition. Assuming no air resistance, find (** a**) an expression for the
displacement of the ball at any time

**, and (**

*t***) the position of the ball at**

*b**t*= π / 2.

**Ans. **ft
4
1)

12 ( );4cos(

2
1)( −=−= π*ytty *

**
Q2. **A 10 kg mass is attached to a spring, stretching it 0.7 m from its natural length. The mass is
started in motion from the equilibrium position, with an initial velocity of 1 m/sec. in the upward
direction. Find the displacement if the force due to air resistance (damping force) is – 90*y′*.

**Ans. **0lim ),(
5
1)( 2772

2 1 =−=+= ∞→

−−−− *yeeececty t
tttt *

**
Q3. **Refer to **Q2**, if the applied external force *F*(*t*) = 5sin*t*, find the displacement at any time *t*.

**Ans. **)cos9sin139990(
500

1 72 *tteey tt *−++−= −−

**
Q4. **Recall that, the **Newton’s Law of Cooling** states that the temperature *T* of an object changes
at a rate proportional to the difference between the temperatures of the object (*T*) and the

surrounding medium (*Ts*). That is, 0)0()( *TTTTkdt
dT
*

*s *=−= . Using this law, solve the following

problems:
**A. **An object takes 40 minutes to cool from 30oC to 24oC in a room having the room-

temperature of 20oC. Find the temperature of the object at any time *t*. What will be the
temperature of the object after 15 minutes? How long will it take the object to cool down to
21oC?

**Hint: ** the IVP is 24)40( ,30)0(),20( ==−= *TTTk
dt
dT *,

**Ans.
**

minutes 5.100 time the,21 when and C1.27)15(,1020 040 )4.0ln(

===+= ⎟ ⎠ ⎞

⎜ ⎝ ⎛

*tTTeT
t
*

**B. **One liter of ice-cream at a temperature of –15oC is removed from the deep freezer and
placed in a room where the temperature is 20oC. If after 20 minutes the temperature of the
ice-cream is –10oC, how long will it take the ice-cream to reach a temperature of 0oC?

**Ans. **minutes 63.72 time the,0 when and ,3520 20
7ln6ln

==−= ⎟ ⎠ ⎞

⎜ ⎝ ⎛ −

*tTeT
t
*

**C. ** A thermometer is taken outside from an air-conditioned room where the temperature is
21oC. After 1 minute it reads 27oC and after 2 min. it reads 30oC. (That is, *T*(1) = 27 and
*T*(2) = 30). Find the outside temperature *Ts*.

**Ans. ***Ts *= 33oC

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**Applied Differential Equations **(MT2043)* Assignment 04*** Submission Date: 18/05/2012
**

** Instructor: Dr. Rashid Ali
**

**Q5.** Find the orthogonal trajectories of all the parabolas with vertices at the origin and foci on
the *x*-axis. (**Hint:** here the give family of curves is *y2* = 4*ax*) **Ans. **2*x*2 + *y*2 = C.
**Q6. **Find the orthogonal trajectories for the one-parameter family *y = – x – 1 + C1ex*.
**
Q7. **Find a second order homogenous linear differential equation for which the functions

*xxyxy *ln and 22
2

1 == are solutions. **Hint: ** first find the Wronskian and use it to verify that the
given solutions are linearly independent, and then write down the general solution of the required
differential equation. Find the required differential equation by eliminating the arbitrary constants
from the general solution.
**Q8. ** Use the method of **undetermined coefficients** to solve the following differential equations:

**i. **234212 2
1

3 1

4
3 Ans.129 *xxxcxcyxxy *−+++=−+=′′

**ii. ***xxxx eececyeyyy *42314 5
2 Ans.232 ++==−′−′′ −

**iii. ***xxxx xeececyeyyy *32313 2
1 Ans.232 ++==−′−′′ −

**iv. ***xxeececyxeyyy xxxx *cossin2
2
1 Ans.sin10232 2

3 1 −+++=−=−′−′′

−

**v.
***xxxxx
*

*xxx
*

*xeexexxececy
*

*exeexyyy
*

32 2 73 Ans.

42223

2322 21

32

−−+++++=

+++=−′−′′

**
Q9. ** Use the method of **variation of parameters** to solve the following differential equations:

**i. ***xxexecxcxxexexecxcy
x
*

*eyyy xxxxx
x
*

lnln Ans.2 3121 ++=+−+==+′−′′

**ii. ***xxxcxcyxyy *2sin
12
12cos

6 12sin2cos Ans.2sin4 2221

2 +++==−′′

**iii. **( )*xxxxcxcyxyy *tanseclncossincos Ans.tan 21 +−+==+′′

**iv.
**242

21

2 21

222

3)1( Ans.

)1( given that )1(622)1(

*xxxcxcy
*

*xcxcyxyyxyx c
*++−+=

−+=+=+′−′′+

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