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الكلية كلية العلوم     القسم قسم الكيمياء     المرحلة 2
أستاذ المادة فؤاد حمزة عبد الشريفي       26/01/2017 10:52:36
الاجوبة النموذجية
Group 1: Differential equations :Solve the differential equations. (Do only 4) (40 marks)
1. 2x^2 y^ -y(2x+y)=0
2x^2 dy/dx-y(2x+y)=0 ? 2x^2 dy-y(2x+y)dx=0
M(tx,ty)=?2x?^2 t^2=t^2 M(x,y) homogeneous of degree 2
N(tx,ty)=yt(2xt+yt)=t^2 y(2x+y)=t^2 N(x,y) homogeneous of degree 2
dy/dx=y(2x+y)/(2x^2 )=(2xy+y^2)/(2x^2 )=( y )/( x )+y^2/(2x^2 )
Put v=( y )/( x )
f(v)=v+( v^2)/2
dx/x=dv/(f(v)-v)
dx/x=2dv/v^2
??dx/x=??2dv/v^2
ln?x=(-2 )/v+c ? ln?x=(-2 x )/( y )+c
2. x^2 y^ -2xy^ +2y=4x^3

Then we get (d^2 y)/(dt^2 )+(-2-1) dy/dt+2y=4e^3t ? (d^2 y)/(dt^2 )-3 dy/dt+2y=4e^3t
m^2-3m+2=0 ?(m-1)(m-2)=0 ? m_1=1 and m_2=2
y_h=c_1 e^t+c_2 e^2t
Let y_p=Ae^3t then dy/dt=3Ae^3t and (d^2 y)/(dt^2 )=9Ae^3t

9Ae^3t-9Ae^3t+2Ae^3t=4e^3t ? A=2 ? y_p=2e^3t
So y=y_h+y_p=c_1 e^2t+c_2 e^t+2e^3t
We have x=e^t
Then y=c_1 x^2+c_2 x+2x^3


3.y^ -5y^ +6y=x^2 e^3x
m^2-5m+6=0 ?(m-3)(m-2)=0 ? m_1=3 and m_2=2
y_h=c_1 e^3x+c_2 e^2x
v_1^ (x) e^3x+v_2^ (x) e^2x=0
3v_1^ (x) e^3x+2v_2^ (x) e^2x=x^2 e^3x
v_1^ (x)=|?(0&e^2x@x^2 e^3x&2e^2x )|/|?(e^3x&e^2x@3e^3x&2e^2x )| =(-x^2 e^3x×e^2x)/(e^3x×2e^2x-3e^3x×e^2x )=x^2 ? v_1 (x)=x^3?3
v_2^ (x)=|?(e^3x&0@3e^3x&x^2 e^3x )|/|?(e^3x&e^2x@3e^3x&2e^2x )| =(e^3x×x^2 e^3x)/(e^3x×2e^2x-3e^3x×e^2x )=(x^2 e^6x)/(-e^5x )=-x^2 e^x

v_2 (x)=-x^2 e^x+2xe^x-2e^x=e^x (-x^2+2x-2)
y_p=v_1 y_1+v_2 y_2=(x^3?3) e^3x+e^x (-x^2+2x-2) e^2x=e^3x ((x^3?3)-x^2+2x-2)
y=y_h+y_p=c_1 e^3x+c_2 e^2x+e^3x ((x^3?3)-x^2+2x-2)
4. y^ +9y^ +14y=0 with y^ (5?)=2 and y(5?)=4
m^2+9m+14=0
(m+2)(m+7)=0 ? m_1=-2 and m_2=-7
y_h=c_1 e^(-2x)+c_2 e^(-7x)
y(5?)=4 ? 4=c_1 e^(-10?)+c_2 e^(-35?) ? eq(1)
y_h^ =-2c_1 e^(-2x)-7c_2 e^(-7x)
y^ (5?)=2 ? 2=-2c_1 e^(-10?)-7c_2 e^(-35?) ? eq(2)
eq(1)×2 ? 8=2c_1 e^(-10?)+2c_2 e^(-35?)
Then 10=-5c_2 e^(-35?)
?(c_2=-2e^35? ) ? eq(3)
For eq(3) and eq(1) we get 4=c_1 e^(-10?)-2 ? ?( c_1=6e^10? )
y_h=6e^10? e^(-2x)-2e^35? e^(-7x)=6e^(10?-2x)-2e^(35?-7x)
5. y^ -xy^ +4y=0 by using power series method
y=a_0+a_1 x+a_2 x^2+a_3 x^3+a_4 x^4+?
y^ =a_1+2a_2 x+3a_3 x^2+4a_4 x^3+?
y^ =2a_2+6a_3 x+12a_4 x^2+20a_5 x^3+?
y^ =2a_2+6a_3 x+12a_4 x^2+20a_5 x^3 +30a_6 x^4+?
-xy^ = 0 -a_1 x-2a_2 x^2 -3a_3 x^3 -4a_4 x^4 -?
4y=4a_0 +4a_1 x +4a_2 x^2 +4a_3 x^3 +4a_4 x^4 +?
2a_2+4a_0=0? ?(a_2=-2a_0 ) ,6a_3-a_1+4a_1=0 ? ?(a_3=-(1?2) a_1 )
12a_4-2a_2+4a_2=0 ? a_4=-(1?6) a_2 ? ?(a_4=(1?3) a_0 )
20a_5-3a_3+4a_3=0 ?a_5=-(1?20) a_3 ? ?(a_5=(1?40) a_1 )
30a_6-4a_4+4a_4=0 ? ?(a_6=0) and a_8=a_10=?=0
y=a_0+a_1 x-2a_0 x^2-(1?2) a_1 x^3+(1?3) a_0 x^4+(1?40) a_1 x^5+?
y=a_0 (1-2x^2+x^4/3)+a_1 (x-x^3/2+x^5/40+? )
Group 2: Matrices and Vector functions (30 marks)
6. If A=(?(3&-1@-2&1@1&-3)) and B=(?(2&0&1@1&-2&2)) ,then find det?AB.
AB=(?(3&-1@-2&1@1&-3))(?(2&0&1@1&-2&2))=(?(6-1&0+2&3-2@-4+1&0-2&-2+2@2-3&0+6&1-6))=(?(5&2&1@-3&-2&0@-1&6&-5))
det?AB=|?(5&2&1@-3&-2&0@-1&6&-5)|=3|?(2&1@6&-5)|-2|?(5&1@-1&-5)|=3(-10-6)-2(-25+1)
=-48+48=0










7. Find eigenvalues and eigenvectors for A=(?(-2&-1@3&2)).
?I-A=?(?(1&0@0&1))-(?(-2&-1@3&2))=(?(?&0@0&?))-(?(-2&-1@3&2))=(?(?+2&1@-3&?-2))
|?I-A|= (?-2)(?+2)+3=0
?^2-1=0 ? ?=?1
?=1
(?(?+2&1@-3&?-2))(?(x@y))=(?(0@0)) ? (?(3&1@-3&-1))(?(x@y))=(?(0@0))
3x+y=0 ? y=-3x
X=(?(m@-3m))=m(?(1@-3))
?=-1
(?(?+2&1@-3&?-2))(?(x@y))=(?(0@0)) ? (?(1&1@-3&-3))(?(x@y))=(?(0@0))
x+y=0 ? x=-y
X=(?(-m@m))=m(?(-1@1))
8. If F(x,y,z)=xz sec?y i+y sin?2z j+y cos?3x k then find ?×F
?×F=| ?(i&j&k@?/?x&?/?y&?/?z @xz sec?y&y sin?2z&y cos?3x )|
=(?(y cos?3x )/?y-?(y sin?2z )/?z)i-(?(y cos?3x )/?x-?(xz sec?y )/?z)j +(?(y sin?2z )/?x-?(xz sec?y )/?y)k
=(cos?3x-2y cos?2z )i-(-3y sin?3x-x sec?y )j +(0-xz sec?y tan?y )k
=(cos?3x-2y cos?2z )i+(3y sin?3x+x sec?y )j-(xz sec?y tan?y )k












Group 3: Chemistry applications ( Do only 3) (30 marks)
9. The rate of decomposition of radioactive einsteinium is proportional to the amount
present at any time. The half-life of radioactive radium is 1600 years. If a sample initially contains 30 grams, how much the amount will remain after 250 years?
Let y(t) be the amount of radium present at time t in years.
dy/dt?y ? dy/dt=ky ? dy/y=kdt
??dy/y=??kdt ? ln?y=kt+c ? y=e^(kt+c ) ? y=Ae^(kt )
We have y=30 when t=0 ? 30=Ae^(0 ) ? A=30
Thus y=30e^(kt )
The half-life of radium is 1600 years it s meaning that y=15 when t=1600
15=30e^(1600k ) ? e^(1600k )=0.5 ? 1600k=ln?0.5
k=ln?0.5/1600
So y=30e^(ln?0.5/1600 t )
t=250 ?y=30e^(ln?0.5/1600 ×250 )=26.93 grams
10. A tank contains 50 gallons of a solution composed of 90% water and 10% alcohol. A second solution containing half water and half alcohol is added to the tank at the rate of 5 gallons per minute. At the same time, the tank is being drained at the rate of 5 gallons per minute. Assuming that the solution is stirred constantly, how much alcohol will be in the tank after 10 minutes? by solving the equation.
dy/dt=-5(y/50)+2.5 ? y^ +( 1 )/( 10 ) y=( 5 )/( 2 )
u(t)=e^??? P(t)dt?=e^(??(1?10) dt)=e^(1?10)t
y=1/e^(1?10)t ???( 5 )/( 2 ) e^(1?10)t ? dt ? y=1/e^(1?10)t (25e^(1?10)t+C)
?(y=25+Ce^(-(1?10)t) )
At t=0 we have y=50 ×10%=5
5=25+Ce^(-(1?10)×0) ? C=-20
y=25-20e^(-(1?10)t)
When t=10 then y=25-20e^(-(1?10)×10) ? y=25-20e^(-1)
y=25-7.36=17.64 gallons

11. 5C+10H=70 , 3C+8H+3O=92 and C+4H+O=32
D=|?(5&10&0@3&8&3@1&4&1)|=5|?(8&3@4&1)|-10|?(3&3@1&1)|=-20
D_C=|?(70&10&0@92&8&3@32&4&1)|=70|?(8&3@4&1)|-10|?(92&3@32&1)|=-280+40=-240
D_H=|?(5&70&0@3&92&3@1&32&1)|=5|?(92&3@32&1)|-70|?(3&3@1&1)|=-20
D_O=|?(5&10&70@3&8&92@1&4&32)|=5|?(8&92@4&32)|-10|?(3&92@1&32)|+70|?(3&8@1&4)|=-320
C=D_C/D=(-240)/(-20)=12 The atomic weight of carbon
H=D_H/D=(-20)/(-20)=1 The atomic weight of hydrogen
O=D_O/D=(-320)/(-20)=16 The atomic weight of oxygen
12. Discuss the molecular vibration of CH_3 .
D_3={1,r,r^2,s,sr,sr^2 }

1 is an identity element which make all H constant?(1)
r is a clock wise rotations about the center with angle :(1? 2? 3)
r^2 is a clock wise rotations about the center with angle :(1? 3?2)
s is a reflection about the line L_1 : ( 2?3)
sr is a reflection about the line L_2 :(1?3)
sr^2 is a reflection about the line L_3 :(1?2)


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