اسئلة واجوبة الامتحان الشهري الاول م2 ف2

(1)
1. Solve y^ =9xe^3x with y^ (0)=2 ,y(0)=( 4 )/( 3 )
2. Solve (x^2+x-12)dy=(y^2-3y)dx
3. Solve (2y+tan?2x )dy+(( 1 )/( x )+2y sec^2?2x )dx=0
4. Solve x^2 dy/dx+3xy=cos?3x/x
(2)
1. Solve y^ =4xe^2x with y^ (0)=3/( 2 ) ,y(0)=3

2. Solve (y^2+y-12)dx=(x^2-2x)dy
3. Solve (2x+tan?3y )dx+(( 1 )/( y )+3x sec^2?3y )dy=0
4. Solve x^3 dy/dx+2x^2 y=x sin?2x
(3)
1. Solve y^ =6xe^3x with y^ (0)=( 4 )/( 3 ) ,y(0)=( 5 )/( 9 )
2. Solve (x^2-x-12)dy=(y^2+2y)dx
3. Solve ( 2y+ tan?3x )dy+(( 1 )/( x )+3y sec^2?3x )dx=0
4. Solve x^2 dy/dx+3xy=sin?2x/x
(4)
1. Solve y^ =6xe^2x with y^ (0)=( 3 )/( 2 ) ,y(0)=( 5 )/( 2 )
2. Solve (y^2-y-12)dx=(x^2+3x)dy
3. Solve (3x^2+tan?2y )dx+(( 4 )/( y )+2x sec^2?2y )dy=0
4. Solve x^2 dy/dx+2xy=cos?3x
الاجوبة النموذجية (1)
1. Solve y^ =9xe^3x with y^ (0)=2 ,y(0)=( 4 )/( 3 )
y^ =???9xe^3x dx?=3xe^3x-e^3x+C
y^ (0)=2? 2=0-1+C ?C=3
y^ =3xe^3x-e^3x+3

y=??(3xe^3x-e^3x+3)dx= xe^3x-( 1 )/( 3 ) e^3x -( 1 )/( 3 ) e^3x+3x+K
y(0)=( 4 )/( 3 ) ? ( 4 )/( 3 )=0-( 1 )/( 3 ) -( 1 )/( 3 )+0+K ? K=2
y= xe^3x-( 2 )/( 3 ) e^3x+3x+2
2. Solve (x^2+x-12)dy=(y^2-3y)dx
dy/((y^2-3y) )=dx/((x^2+x-12) ) ? dy/y(y-3) =dx/(x-3)(x+4)
1/y(y-3) =A/y+B/((y-3) ) ; y=0?A=-1/( 3 ) & y=3?B=1/( 3 )
1/y(y-3) =(-1)/3y+1/3(y-3)
1/(x-3)(x+4) =C/((x-3) )+D/((x+4) ) ;x=3 ?C=( 1 )/( 7 ) & x=-4 ?D=-( 1 )/( 7 )
1/(x-3)(x+4) =1/7(x-3) -1/7(x+4)
??((-1)/3y+1/3(y-3) ) dy=??(1/7(x-3) -1/7(x+4) ) dx
1/( 3 ) (-ln?y+ln?(y-1) )=( 1 )/( 7 ) (ln??(x-3)-ln?(x+4) ? )+c


3. Solve (2y+tan?2x )dy+(( 1 )/( x )+2y sec^2?2x )dx=0
?M/?x=2 sec^2?2x , ?N/?y=2 sec^2?2x ? ?N/?x=?M/?y
Which implies that the differential equation is exact.
F=??(2y+tan?2x )dy
F=y^2+y tan?2x+g(x)
Similarly F=??(( 1 )/( x )+2y sec^2?2x )dx
F=ln?x+y tan?2x+?(y)
g(x)=ln?x & ?(y)=y^2
ln?x+y tan?2x+y^2=C

4. Solve x^2 dy/dx+3xy=cos?3x/x
y^ +(( 3 )/x)y=cos?3x/x^3 Standard form, y^ +P(x)y=Q(x)
P(x)=( 3 )/x and Q(x)=cos?3x/x^3 .So,the integrating factor is
u(x)=e^??? P(x)dx?=e^(??3?x dx) =e^(3 ln?x )=e^ln??x^3 ? =x^3
y=1/u(x) ???u(x) ? Q(x)dx
y=1/x^3 ??x^3 ×cos?3x/x^3 dx=1/x^3 ??cos?3x dx
y=1/x^3 (( 1 )/( 3 ) sin?3x+c)








الاجوبة النموذجية (2)

1. Solve y^ =4xe^2x with y^ (0)=3/( 2 ) ,y(0)=3
y^ =???4xe^2x dx?=2xe^2x-e^2x+C
y^ (0)=3/( 2 ) ?3/( 2 )=-1+C ? C=5/( 2 )
y^ =2xe^2x-e^2x+5/( 2 )
y=??(2xe^2x-e^2x+5/( 2 ))dx= xe^2x-( 1 )/( 2 ) e^x -( 1 )/( 2 ) e^2x+5/( 2 ) x+K
y(0)=3 ? 3=0-( 1 )/( 2 ) -( 1 )/( 2 )+0+K ? K=4
y= xe^2x- e^x+5/( 2 ) x +4

2. Solve (y^2+y-12)dx=(x^2-2x)dy
dy/((y^2+y-12) )=dx/((x^2-2x) ) ? dy/(y-3)(y+4) =dx/x(x-2)
1/x(x-2) =A/x+B/((x-2) ) ; x=0?A=-1/( 2 ) & x=2?B=1/( 2 )
1/x(x-2) =(-1)/2x+1/2(x-2)
1/(y-3)(y+4) =C/((y-3) )+D/((y+4) ) ;y=3 ?C=( 1 )/( 7 ) & y=-4 ?D=-( 1 )/( 7 )
1/(y-3)(y+4) =1/7(y-3) -1/7(y+4)
??((-1)/2x+1/2(x-2) ) dx=??(1/7(y-3) -1/7(y+4) ) dy
1/( 2 ) (-ln?x+ln?(x-2) )=( 1 )/( 7 ) (ln??(y-3)-ln?(y+4) ? )+c



3. Solve (2x+tan?3y )dx+(( 1 )/( y )+3x sec^2?3y )dy=0
?M/?y=3 sec^2?3y , ?N/?x=3 sec^2?3y ? ?N/?x=?M/?y
Which implies that the differential equation is exact.
F=??(2x+tan?3y )dx
F=x^2+x tan?3y+?(y)
Similarly F=??(( 1 )/( y )+3x sec^2?3y )dy
F=ln?y+x tan?3y+g(x)
g(x)=x^2 & ?(y)=ln?y
x^2+x tan?3y+ln?y=C

4. Solve x^3 dy/dx+2x^2 y=x sin?2x
y^ +(( 2 )/x)y=sin?2x/x^2 Standard form, y^ +P(x)y=Q(x)
P(x)=( 2 )/x and Q(x)=sin?2x/x^2 .So,the integrating factor is
u(x)=e^??? P(x)dx?=e^(??2?x dx) =e^(2 ln?x )=e^ln??x^2 ? =x^2
y=1/u(x) ???u(x) ? Q(x)dx
y=1/x^2 ??x^2 ×sin?2x/x^2 dx=1/x^2 ??sin?2x dx
y=1/x^2 (-( 1 )/( 2 ) cos?2x+c)






الاجوبة النموذجية (3)
1. Solve y^ =6xe^3x with y^ (0)=( 4 )/( 3 ) ,y(0)=( 5 )/( 9 )

y^ =???6xe^3x dx?=2xe^3x-( 2 )/( 3 ) e^3x+C

y^ (0)=( 4 )/( 3 )? ( 4 )/( 3 )=0-( 2 )/( 3 )+C ?C=2
y^ =2xe^3x-( 2 )/( 3 ) e^3x+2
y=??(2xe^3x-( 2 )/( 3 ) e^3x+2)dx= ( 2 )/( 3 ) xe^3x-( 2 )/( 9 ) e^3x -( 2 )/( 9 ) e^3x+2x+K
y(0)=( 5 )/( 9 ) ? ( 5 )/( 9 )=0-( 2 )/( 9 ) -( 2 )/( 9 )+0+K ? K=1
y= ( 2 )/( 3 ) xe^3x-( 4 )/( 9 ) e^3x+2x+1

2. Solve (x^2-x-12)dy=(y^2+2y)dx
dy/((y^2+2y) )=dx/((x^2-x-12) ) ? dy/y(y+2) =dx/(x+3)(x-4)
1/y(y+2) =A/y+B/((y+2) ) ; y=0?A=( 1 )/2 & y=-2?B=-( 1 )/2
1/y(y+2) =( 1 )/2y-1/2(y+2)
1/(x+3)(x-4) =C/((x+3) )+D/((x-4) ) ;x=-3 ?C=-( 1 )/( 7 ) & x=4 ?D=( 1 )/( 7 )
1/(x+3)(x-4) =1/7(x-4) -1/7(x+3)
??(1/2y-1/2(y+2) ) dy=??(1/7(x-4) -1/7(x+3) ) dx
( 1 )/2 (ln?y-ln?(y+2) )=( 1 )/( 7 ) (ln??(x-4)-ln?(x+3) ? )+c

3. Solve ( 2y+ tan?3x )dy+(( 1 )/( x )+3y sec^2?3x )dx=0
?M/?x=3 sec^2?3x , ?N/?y=3 sec^2?3x ? ?N/?x=?M/?y
Which implies that the differential equation is exact.
F=??(( 1 )/( x )+3y sec^2?3x )dx
F=ln?x+y tan?3x+?(y)
Similarly F=??(2y+ tan?3x )dy
F=y^2+y tan?3x+g(x)
g(x)=ln?x & ?(y)=y^2
y^2+y tan?3x+ln?x=C

4. Solve x^2 dy/dx+3xy=sin?2x/x
y^ +(( 3 )/x)y=sin?2x/x^3 Standard form, y^ +P(x)y=Q(x)
P(x)=( 3 )/x and Q(x)=sin?2x/x^3 .So,the integrating factor is
u(x)=e^??? P(x)dx?=e^(??3?x dx) =e^(3 ln?x )=e^ln??x^3 ? =x^3
y=1/u(x) ???u(x) ? Q(x)dx
y=1/x^3 ??x^3 ×sin?2x/x^3 dx=1/x^3 ??sin?2x dx
y=1/x^3 (-( 1 )/( 2 ) cos?2x+c)






الاجوبة النموذجية (4)
1. Solve y^ =6xe^2x with y^ (0)=( 3 )/( 2 ) ,y(0)=( 5 )/( 2 )

y^ =???6xe^2x dx?=3xe^2x-( 3 )/( 2 ) e^2x+C

y^ (0)=( 3 )/( 2 )? 3=0-( 3 )/( 2 )+C ?C=3
y^ =3xe^2x-( 3 )/( 2 ) e^2x+3
y=??(3xe^2x-( 3 )/( 2 ) e^2x+3)dx= ( 3 )/( 2 ) xe^2x-( 3 )/( 4 ) e^2x -( 3 )/( 4 ) e^2x+3x+K
y(0)=( 5 )/( 2 ) ? ( 5 )/( 2 )=0-( 3 )/( 4 ) -( 3 )/( 4 )+0+K ? K=4
y= ( 3 )/( 2 ) xe^2x-( 3 )/( 2 ) e^2x +3x+4

2. Solve (y^2-y-12)dx=(x^2+3x)dy
dy/((y^2-y-12) )=dx/((x^2+3x) ) ? dy/(y+3)(y-4) =dx/x(x+3)
1/x(x+3) =A/x+B/((x+3) ) ; x=0?A=( 1 )/3 & x=-3?B=-( 1 )/3
1/x(x-3) =1/3x-1/3(x-3)
1/(y+3)(y-4) =C/((y+3) )+D/((y-4) ) ;y=-3 ?C=-( 1 )/( 7 ) & y=4 ?D=( 1 )/( 7 )
1/(y+3)(y-4) =1/7(y-4) -1/7(y+3)
??(1/3x-1/((x-3) )) dx=??(1/7(y-4) -1/7(y+3) ) dy
1/( 3 ) (ln?x-ln?(x-3) )=( 1 )/( 7 ) (ln??(y-4)-ln?(y+3) ? )+c

3. Solve (3x^2+tan?2y )dx+(( 4 )/( y )+2x sec^2?2y )dy=0
?M/?y=2 sec^2?2y , ?N/?x=2 sec^2?2y ? ?N/?x=?M/?y
Which implies that the differential equation is exact.
F=??(3x^2+tan?2y )dx
F=x^3+x tan?2y+?(y)
Similarly F=??(( 4 )/( y )+2x sec^2?2y )dy
F=4 ln?y+x tan?2y+g(x)
g(x)=x^3 & ?(y)=4 ln?y
x^3+x tan?2y+4 ln?y=C

4. Solve x^2 dy/dx+2xy=cos?3x
y^ +(( 2 )/x)y=cos?3x/x^2 Standard form, y^ +P(x)y=Q(x)
P(x)=( 2 )/x and Q(x)=cos?3x/x^2 .So,the integrating factor is
u(x)=e^??? P(x)dx?=e^(??2?x dx) =e^(2 ln?x )=e^ln??x^2 ? =x^2
y=1/u(x) ???u(x) ? Q(x)dx
y=1/x^2 ??x^2 ×cos?3x/x^2 dx=1/x^2 ??cos?3x dx
y=1/x^2 (( 1 )/( 3 ) sin?3x+c)