Differential Equations Civil Engineering (CE) Notes | EduRev

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Civil Engineering (CE) : Differential Equations Civil Engineering (CE) Notes | EduRev

The document Differential Equations Civil Engineering (CE) Notes | EduRev is a part of the Civil Engineering (CE) Course Topic wise GATE Past Year Papers for Civil Engineering.
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Q.1 An ordinary differential equation is given below.
Differential Equations Civil Engineering (CE) Notes | EduRev
The solution for the above equation is
(Note: K denotes a constant in the options)
(a) y = Kxex
(b) y = Kxe-x
(c) y = Klnx
(d) y = Kxlnx     [2019 : 2 Marks, Set-II]
Ans:
(c)

Differential Equations Civil Engineering (CE) Notes | EduRev
Differential Equations Civil Engineering (CE) Notes | EduRev

Q. 2 Consider the ordinary differential equation  Differential Equations Civil Engineering (CE) Notes | EduRev Given the values of y(1) = 0 and y(2) = 2, the value of y{3) (round off to 1 decimal place), is_______     [2019 : 2 Marks, Set-I]
Ans: y{3} = (6)


Q. 3 A one-dimensionai domain is discretized into N sub-domains of width Δx with node numbers i = 0,1,2, 3 ....... N. If the time scale is discretized in steps of Δt, the forward-time and centered- space finite difference approximation at nth node and time step, for the partial differential equation Differential Equations Civil Engineering (CE) Notes | EduRev

(a) Differential Equations Civil Engineering (CE) Notes | EduRev

(b) Differential Equations Civil Engineering (CE) Notes | EduRev

(c) Differential Equations Civil Engineering (CE) Notes | EduRev

(d) Differential Equations Civil Engineering (CE) Notes | EduRev            [2019 : 2 Marks, Set-I]

Ans: (b)

Differential Equations Civil Engineering (CE) Notes | EduRev

Approximate time derivative in equation (i) with forward difference,
Differential Equations Civil Engineering (CE) Notes | EduRev
Note that the right hand side only in value v at x = xi 
Use the central difference approximation to Differential Equations Civil Engineering (CE) Notes | EduRev and evaluate all the terms at time n.

Differential Equations Civil Engineering (CE) Notes | EduRev

Substituting equation (ii) in the left hand side of equation (i), substitute the equation (iii) into the right hand side of equation (i), and collect the truncation error terms to get
Differential Equations Civil Engineering (CE) Notes | EduRev


Q. 4 The solution of the equation Differential Equations Civil Engineering (CE) Notes | EduRev passing through the point (1,1) is

(a) x
(b) x2
(c) x-
(d) x-2     [2018 : 1 Mark, Set-II]

Ans: (c)

Differential Equations Civil Engineering (CE) Notes | EduRev
Differential Equations Civil Engineering (CE) Notes | EduRev


Q. 5 The solution (up to three decimal places) at x = 1 of the differential equation Differential Equations Civil Engineering (CE) Notes | EduRev subject to boundary conditions y(0) = 1 and 

Differential Equations Civil Engineering (CE) Notes | EduRev          [2018 : 2 Marks, Set-I]
Ans: 0.368

Differential Equations Civil Engineering (CE) Notes | EduRev
From eq. (ii) and (iii),
Differential Equations Civil Engineering (CE) Notes | EduRev


Q. 6 Consider the following second-order differential equation: 
y "- 4y' + 3 y = 2t - 3t2

The particular solution of the differential equation is 
(a) - 2 - 2t - t2 
(b) - 2t - t2 
(c) 2t - t2 
(d) - 2 - 2t - 3t2       [2017 : 2 Marks, Set-II]
Ans: (a)
Differential Equations Civil Engineering (CE) Notes | EduRev
Differential Equations Civil Engineering (CE) Notes | EduRev
Differential Equations Civil Engineering (CE) Notes | EduRev


Q. 7 The solution of the equation Differential Equations Civil Engineering (CE) Notes | EduRev with Q = 0 at t = 0 is
(a) Q(t) = e-t - 1
(b) Q(t) = 1 + e-t
(c) Q(t) = 1 - et

(d) Q(f) = 1 - e-t        [2017 : 2 Marks, Set-I]
Ans : (d)

Differential Equations Civil Engineering (CE) Notes | EduRev
comparing with standard form
Differential Equations Civil Engineering (CE) Notes | EduRev


Q. 8 Consider the following partial differential equation:
Differential Equations Civil Engineering (CE) Notes | EduRev
For this equation to be classified as parabolic, the value of B2 must be ________ .       [2017 : 1 Mark, Set-I]
Ans: 36

Given that the partial differential equation is parabolic.
Differential Equations Civil Engineering (CE) Notes | EduRev


Q. 9 The respective expressions for complimentary function and particular integral part of the solution of the differential equation
Differential Equations Civil Engineering (CE) Notes | EduRev
Differential Equations Civil Engineering (CE) Notes | EduRev
Differential Equations Civil Engineering (CE) Notes | EduRev
Ans: a

Differential Equations Civil Engineering (CE) Notes | EduRev
Differential Equations Civil Engineering (CE) Notes | EduRev
Differential Equations Civil Engineering (CE) Notes | EduRev


Q. 10 The solution of the partial differential equation Differential Equations Civil Engineering (CE) Notes | EduRev is of the form 
Differential Equations Civil Engineering (CE) Notes | EduRev
Differential Equations Civil Engineering (CE) Notes | EduRev
Ans: (b)

Differential Equations Civil Engineering (CE) Notes | EduRev

Differential Equations Civil Engineering (CE) Notes | EduRev
Differential Equations Civil Engineering (CE) Notes | EduRev
Differential Equations Civil Engineering (CE) Notes | EduRev


Q. 11 The type of partial differential equation
Differential Equations Civil Engineering (CE) Notes | EduRev

(a) elliptic
(b) parabolic
(c) hyperbolic
(d) none of these    [2016 : 1 Mark, Set-I]
Ans: (c)
Comparing the given equation with the general form of second order partial differential equation, we have A = t , B = 3, C = 1
=> B2 - 4AC = 5 > 0
∴ PDE is Hyperbola.


Q. 12 Consider the following second order linear differential equation
Differential Equations Civil Engineering (CE) Notes | EduRev
The boundary conditions are: at x = 0, y = 5 and x = 2, y = 21
The value of y at x = 1 is .    [2015 : 2 Marks, Set-II]

Ans: y = 18

Differential Equations Civil Engineering (CE) Notes | EduRev
Integrating both sides wrt. x,
Differential Equations Civil Engineering (CE) Notes | EduRev
Integrating both sides wrt. x
Differential Equations Civil Engineering (CE) Notes | EduRev

Differential Equations Civil Engineering (CE) Notes | EduRev


Q. 13 Consider the following difference equation
Differential Equations Civil Engineering (CE) Notes | EduRev
Which of the following is the solution of the above equation (c is an arbitrary constant)?          [2015 : 2 Marks, Set-I]
(a) Differential Equations Civil Engineering (CE) Notes | EduRev
(b) Differential Equations Civil Engineering (CE) Notes | EduRev
(c) Differential Equations Civil Engineering (CE) Notes | EduRev
(d) Differential Equations Civil Engineering (CE) Notes | EduRev       
Ans: c

Differential Equations Civil Engineering (CE) Notes | EduRev
Differential Equations Civil Engineering (CE) Notes | EduRev
Integrating both side
Differential Equations Civil Engineering (CE) Notes | EduRev


Q. 14 The integrating factor for differential equation 
Differential Equations Civil Engineering (CE) Notes | EduRev  [2014 : 1 Mark, Set-II]
(a) Differential Equations Civil Engineering (CE) Notes | EduRev
(b) Differential Equations Civil Engineering (CE) Notes | EduRev
(c) Differential Equations Civil Engineering (CE) Notes | EduRev
(d) Differential Equations Civil Engineering (CE) Notes | EduRev    

Ans: (d)


Q. 15 The solution of the ordinary differential equation Differential Equations Civil Engineering (CE) Notes | EduRev for the boundary condition, y = 5 at x = 1 is     [2011 : 2 Marks]
(a) y = e-2x
(b) y = 2e-2x
(c) y = 10.95 e-2x 
(d) y = 36.95 e-2x

Ans: (d)

Given
Differential Equations Civil Engineering (CE) Notes | EduRev

Differential Equations Civil Engineering (CE) Notes | EduRev

Differential Equations Civil Engineering (CE) Notes | EduRev


Q. 16 The solution of the diffrential equation  Differential Equations Civil Engineering (CE) Notes | EduRev with the condition that y = 1 at x = 1 is    [2011 : 2 Marks
Differential Equations Civil Engineering (CE) Notes | EduRev
Differential Equations Civil Engineering (CE) Notes | EduRev
Differential Equations Civil Engineering (CE) Notes | EduRev
Differential Equations Civil Engineering (CE) Notes | EduRev
Ans: (d)

Differential Equations Civil Engineering (CE) Notes | EduRev

This is a linear differential equation
Differential Equations Civil Engineering (CE) Notes | EduRev
Differential Equations Civil Engineering (CE) Notes | EduRev
solution is
Differential Equations Civil Engineering (CE) Notes | EduRev
Differential Equations Civil Engineering (CE) Notes | EduRev
Differential Equations Civil Engineering (CE) Notes | EduRev


Q. 17 A parabolic cable is held between two supports at the same level. The horizontal span between the supports is L. The sag at the mid-span is h. The equation of the parabola is Differential Equations Civil Engineering (CE) Notes | EduRev where x is the horizontal coordinate and y is the vertical coordinate with the origin at the centre of the cable. The expression for the total length of the cable is
[2010 : 2 Marks]

Differential Equations Civil Engineering (CE) Notes | EduRev
Differential Equations Civil Engineering (CE) Notes | EduRev
Differential Equations Civil Engineering (CE) Notes | EduRev
Differential Equations Civil Engineering (CE) Notes | EduRev
Ans: (d)

Length of curve f(x) between x = a and x = b is given by,
Differential Equations Civil Engineering (CE) Notes | EduRev
Differential Equations Civil Engineering (CE) Notes | EduRev
since, y = 0 at x= 0
and y = h at x = L/2
(As can be seen from equation (i), by substituting x = 0 and x = L/2)
Differential Equations Civil Engineering (CE) Notes | EduRev


Q. 18 The partial differential equation that can be formed from z = ax + by + ab has the form (with Differential Equations Civil Engineering (CE) Notes | EduRev    [2010 : 2 Marks]

(a) z = px + qy
(b) z = px + pq
(c) z = px + qy + pq
(d) z = qy + pq

Ans: (c)

z = a x + b y + a b ...(i)
Differential Equations Civil Engineering (CE) Notes | EduRev
Substituting a and b in (i) in terms of p and q, we get,
z = px + qy + pq


Q. 19 The solution to the ordinary differential equation
Differential Equations Civil Engineering (CE) Notes | EduRev       [2010 : 2 Marks]
(a) y= c1e3x + c2e-2x
(b) y= c1e3x + c2e2x
(c) y= c1e-3x + c2e2x
(d) y= c1e-3x + c2e-2x
Ans: (c) 

Differential Equations Civil Engineering (CE) Notes | EduRev
∴ Solution is y = = c1e-3x + c2e2x


Q. 20 The order and degree of the differential equation Differential Equations Civil Engineering (CE) Notes | EduRev are respectively    [2010 : 1 Mark]
(a) 3 and 2
(b) 2 and 3
(c) 3 and 3
(d) 3 and 1 
Ans (a) 

Differential Equations Civil Engineering (CE) Notes | EduRev
Removing radicals we get,
Differential Equations Civil Engineering (CE) Notes | EduRev
∴ The order is 3 since highest differential is Differential Equations Civil Engineering (CE) Notes | EduRev
The degree is 2 since power of highest differential is 2.



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