Differential Equations Answers

8 One of these is a homogeneous equation

I. h(x,y)=x^3+2xy+3xy^2+4y^3

II. h(x,y)=x^2+2x^2y+3xy^2+4y^3

III. h(x,y)=x^3+2x^2y+3xy^2+4y^3

IV. h(x,y)=x^3+2xy+3xy^2+4y^2
9 One of the following is not a separable equation

i. dy/dx=e^x+y

II. dy/dx=e^xy

III. dy/dx=x^2(y+y^2)

IV. (1+y^2)dx+(1+x^3)dy

4 Find the total differential of the function u = x^2y−3y
I. 2x+(x^2−3)dy
II. 2xydx+x^2dy
III. 2xydx+(x^2−3)dy
IV. 2xydx+(x^3−2)dy
5 The total differential du of a function u (x, y) = 0 is defined as
I. ∂u/∂x dx+∂u.∂x dy=0
II. ∂u/∂x dy+∂u/∂x dy=0
III. ∂u/∂x dx+∂u/∂y dx=0
IV. ∂u/∂x dx+∂u/∂y dy=0
6 A differential equation involving only a single independent variable is called ………… equation.
I. extraordinary differential
II. ordinary differential
III. super-ordinary differential
IV. partial differential
7 The homogeneous equation d/ydx=x^4+x^3y+y^4/3x^3y+y^4 is of …… degree
I. first , II. second , III. third , IV. fourth

1 The order of a differential equation is the
I. order of the highest order derivative appearing in the equation
II. order of the lowest order derivative appearing in the equation
III. order of the second highest order derivative appearing in the equation
Iv. last order of the highest order derivative appearing in the equation
2 If u = f(x,y) be a function of two independent variables x and y, then ∂u/∂y is equal to

i. lim△x→0f(x+△x,y)−f(x,y)/△x

II. lim△y→0f(x,y+△y)−f(x,y)/△y

iii.limy→0f(x,y+△y)−f(x,y)/△y

IV.lim△y→0f(x+△x,y)−f(x,y)/△x
3 The general solution of a first order differential equation normally contains one arbitrary constant which is called a …………..
I. family of curves

ii. perimeter

III. curve

IV. parameter

6 If A=5t^2+tj−t^3k and B=sinti−costj. evaluate d/dt( A×B)
7 If A=5t^2+tj−t^3k and B=sinti − costj. evaluate d/dt (A⋅A)
8 If A= sin ui + cos uj + uk, B = cos ui − sin uj − 3k and C=2i+3j−k, evaluate d/du (A×(B×C)) at u=0
9 Let A=x^2yzi−2xz^3j−xz^2 and B=4zi+yj+4x^2k, find
∂^2/∂x∂y (A×B) at (1,0,-2)
10 solve
d^2A/dt^2 − 4dA/dt − 5A = 0

1 Given that A=sinti+costj+ tk, evaluate ∣d^2A/dt^2∣
2 A particle moves along a curve whose parameter equations are x=e^−t, y=2cos3t, z=2sin3t. Find the magnitude of the acceleration at t=0
3 A particle moves along the curve x=2t^2, y=t^2−4t and z=3t−5,where t is the time. Find the components of the velocity at t=1 in the direction i−3j+2k
4 Determine the unit tangent at the point where t=2 on the curve x= t^2+1,y= 4t−3 and z=2t^2−6t
5 If A=5t^2+tj−t^3k and B=sinti−costj. evaluate d/dt (A⋅B)

7 Obtain the differential equation associated with the given primitive
lny=Ax^2 + B, A and B being arbitrary constants.
(I) xy(d^2y/dx^2) – y dy/dx − x(dy/dx)^2=0
(II) xy (d^2y/dx^2) − 2y dy/dx=0
(III) 3xy (d^2y/dx^2) + 2y dy/dx − x(dy/dx)^2 =0
(IV) Xy (d^2y/dx^2) − x(dy/dx)^2=0
8 Solve y(xy+1)dx+x(1+xy+x^2y^2)dy=0
(I) 3x^2y^2lny−2xy−3=Cx^2y^2
(II) 2x^2y^2lny−2xy−1=Cx^2y^2
(III) 2x^2y^2lny−xy=Cx^2y^2
(IV) 2x^3y^2lny−2xy−1=Cx^2y^5

3 Derive the differential equation associated with the primitive
y=Ax^3+Bx^2 + Cx + D
where A, B, C and D are arbitrary constants.

(a) D^3y/dx^2= 0
(b) d^4y/dx^4 + d^3y/dx^3 = 0
( c) d^3y/dx^3 + d^2y/dx^2 = 0
(d) d^4y/dx^4 = 0
5 Derive the differential equation for the area bounded by the arc of a curve, the x- axis, and the two ordinates, one fixed and one variable, is equal to trice the length of the arc between the ordinates
(I) y=2√4+ (dx/dy)^2
(II) Y =√1+(d^2y/dx^2)^2
(III) Y =2 √1+ (dy/dx)^2
(IV) y=3 √2+(dy/dx)^2
6 Find the differential equation of all straight lines at a unit distance from the origin
(i) (x dy/dx − y)^2=1/2)^2
(II) (x dy/dx − y)^2 =1+ (dy/dx)^2
(III) (3x dy/dx − y)^2=3+(dy/dx)^2
(IV) (2x d^2y/dx^2 − y)^2=1+(dy/dx)^2

1 H grams of artificial sugar in water are being converted into dextrose at a rate which is proportional to the square of the amount unconverted. Find the differential equation expressing the rate of conversion after v minutes given that s grams is converted in v minutes and c being the constant of proportionality.
(a)ds/dv = c(H−s)2
(b) ds/dv = c(H−s)^2
© ds/dv/= (s−H)^2
(d) dv/ds/= c(H−s)
2 A vehicle of mass m moves along a straight line ( the – axis) while subject to a force indirectly proportional to its displacement x from a fixed point O in its path and 2) a resisting force proportional to its acceleration. Express the total force as a differential equation.

(a) 7md^2/xdt^2=k1/t − k2dx/dt
(b) Md^2x/dt^2=−k1/t−k2d^2x/dt^2
© m d^2x/dt^2=−k1/x−k2d^2x/dt^2
(d) M d^−2x/dt^2=−2 k1/x − k2 d^2x/dt^2

8 Solve: y(xy+1) dx + x(1+xy+x^2y^2) dy=0
(a) 2x^2y^2lny − 2xy −1=Cx^2y^2
(b) 3x^2y^2 lny − 2xy −3=Cx^2y^2
© 2x^2y^2 lny − xy=Cx^2y^2
(d) 2x^3y^2lny −2xy −1=Cx^2y^5
9 Solve
Xdy – ydx –√ x^2−y^2 dx = 0
(a) Cx = 2e^arcsiny/x
(b) Cx = e^arcsiny/x
© Cx = e^arcsin2y/3x
(d) Cx = e^arccosy/x
10 The population of student P at NOUN increases at a rate proportional to the population and to the addition of 150,250 and the population divided by 3, the differential equation of this statement is
(a) dP/dT=3 kP(150,250+P)/4
(b) dP/dT = 2kP(150,250+P)/3
© 5 dP/dT = kP(150,250+P)/3
(d) dP/dT=kP(150,250+P)/3

5 Solve the variable separable x^3dx+(y+1)^2dy=0
(a) 3x^4+4(y+2)^3=C
(b) 5x^4+4(y+1)^3=C
© 3x^4+4(y+1)^3=C
(d) 4x^4+4(y+2)^3=C
6 Solve : (x^3+y^3)dx−3xy^2dy=0
(a) x^5−2y^3=Cx
(b) x^3−2y^3=Cx
© x^3−3y^3=Cx
(d) x^3−2y^2=Cx
7 Solve : (1+2e^x/y)dx+2e^x/y(1− x/y)dy=0
(a) 5x+2ye^x/y=C
(b) x+2ye^2x/y=C
© x+2ye^x/y=C
(d) 5x+3ye^x/y=C