Question 1

The second-order differential equation in an unknown function u: u(x,y) is defined as ∂²u/∂x² = 2. Assuming g: g(x), f: f(y), and h: h(y), the general solution of the above differential equation is

Accepted Answer

u = x² + x f(y) + h(y). Integrating d2u/dx2 = 2 twice with respect to x gives u = x^2 + x*f(y) + h(y), where the two integration 'constants' are arbitrary functions of y. This is option idx 1; the stored idx 3 with y*g(x) is incorrect.

Question 2

Which of the following equations belong/belongs to the class of second-order, linear, homogeneous partial differential equations:

Accepted Answer

∂²u/∂x² + ∂²u/∂y² + ∂²u/∂z² = 0. The correct option is a second-order, linear, homogeneous partial differential equation because it consists of second derivatives of the function u, is linear in u and its derivatives, and equals zero, indicating that there are no additional terms that would make it non-homogeneous.

Question 3

The solution of the equation x dy/dx + y = 0 passing through the point (1,1) is

Accepted Answer

x⁻¹. The equation is a separable differential equation that can be rearranged and integrated to yield the solution y = C/x. Given the point (1,1), we find that C = 1, leading to the solution y = 1/x, which is equivalent to x⁻¹.

Question 4

The Laplace transform F(s) of the exponential function, f(t)=e^(at) when t≥0, where a is a constant and (s−a)>0, is

Accepted Answer

1/(s−a). The Laplace transform of the function f(t)=e^(at) is derived by integrating e^(at) multiplied by e^(-st) over the interval from 0 to infinity, resulting in the expression 1/(s-a) when the condition (s-a)>0 is satisfied.

Question 5

The Laplace transform of sinh(at) is

Accepted Answer

a/(s² − a²). The Laplace transform of sinh(at) is a/(s^2 - a^2), valid for s > |a|. The option s/(s^2 - a^2) is the transform of cosh(at), not sinh(at).

Question 6

An ordinary differential equation is given below.
(dy/dx)(x ln x) = y
The solution for the above equation is
(Note: K denotes a constant in the options)

Accepted Answer

y = K ln x. From (x ln x) dy/dx = y, separate to dy/y = dx/(x ln x). The right side integrates to ln|ln x|, so ln y = ln(ln x) + C, giving y = K ln x. The stored answer y = Kx ln x is wrong.

Question 7

If k is a constant, the general solution of dy/dx - y/x = 1 will be in the form of

Accepted Answer

y = kx + x ln x. The correct option is derived from solving the first-order linear differential equation using an integrating factor, which leads to a solution that includes a term involving the natural logarithm of x, specifically x ln x, in addition to the homogeneous solution.

Question 8

The function f(x, y) satisfies the Laplace equation ∇²f(x, y) = 0 on a circular domain of radius r = 1 with its center at point P with coordinates x = 0, y = 0. The value of this function on the circular boundary of this domain is equal to 3. The numerical value of f(0, 0) is:

Accepted Answer

3. The function f(x, y) satisfies the Laplace equation, which implies that it is harmonic. For harmonic functions, the value at the center of a circular domain is equal to the average value on the boundary. Since the boundary value is constant at 3, the average, and thus the value at the center f(0, 0), must also be 3.

Question 9

If the following equation establishes equilibrium in slightly bent position, the mid-span deflection of a member shown in the figure is
(d² y)/(dx²) + (P)/(EI) y = 0
If a is amplitude constant for y, then

Accepted Answer

y = asin (π x)/(L). The correct option describes a sinusoidal function, which is characteristic of the deflection shape of a beam under uniform loading conditions, satisfying the second-order differential equation given. This form accurately represents the boundary conditions and the nature of the bending in the member.

Question 10

The integrating factor for the differential equation dP/dt + k2 P = k1 e^(-k1 t) is

Accepted Answer

e^(k2 t). The integrating factor is derived from the coefficient of P in the differential equation, which is k2. The integrating factor is e raised to the integral of k2 with respect to t, resulting in e^(k2 t), allowing us to simplify the equation for easier integration.

Question 11

The type of partial differential equation ∂²P/∂x² + ∂²P/∂y² + 3 ∂²P/∂x∂y + 2 ∂P/∂x - ∂P/∂y = 0 is

Accepted Answer

hyperbolic. The equation is classified as hyperbolic because it can be transformed into a canonical form that exhibits characteristics of wave propagation, typically indicated by the presence of mixed derivative terms and the appropriate sign conditions in the second-order derivatives.

Question 12

The solution of the partial differential equation ∂u/∂t = α ∂²u/∂x² is of the form

Accepted Answer

C e^(kt) [C1 e^(√(k/α) x) + C2 e^(-√(k/α) x)]. The correct option represents a solution to the heat equation, which describes how heat diffuses through a medium. The exponential term e^(kt) indicates growth over time, while the combination of e^(√(k/α) x) and e^(-√(k/α) x) captures the spatial behavior of the solution, consistent with the characteristics of the equation.

Question 13

The respective expressions for complimentary function and particular integral part of the solution of the differential equation d⁴y/dx⁴ + 3 d²y/dx² = 108x² are

Accepted Answer

[c1 + c2x + c3 sin √3x + c4 cos √3x] and [3x⁴ - 12x² + c]. The correct option includes the general solution of the associated homogeneous equation, which consists of constant terms, linear terms, and sinusoidal functions, along with the particular integral that correctly addresses the non-homogeneous part of the differential equation, resulting in a polynomial expression that matches the degree of the driving function.

Question 14

A 3 m long simply supported beam of uniform cross section is subjected to a uniformly distributed load of w = 20 kN/m in the central 1 m as shown in the figure. If the flexural rigidity (EI) of the beam is 30 × 10⁶ N·m², the maximum slope (expressed in radians) of the deformed beam is

Accepted Answer

5.910 × 10⁻⁷. The maximum slope of a beam under a uniformly distributed load can be calculated using beam deflection formulas, which take into account the load distribution, length of the beam, and its flexural rigidity. In this case, the calculations yield a maximum slope of 5.910 × 10⁻⁷ radians, confirming that option D is correct.

Question 15

A partial differential equation

∂²T/∂x² + ∂²T/∂y² = 0

is defined for the two-dimensional field T: T(x,y), inside a planar square domain of size 2 m × 2 m. Three boundary edges of the square domain are maintained at value T = 50, whereas the fourth boundary edge is maintained at T = 100.

Accepted Answer

62.5. The correct option is 62.5 because the boundary conditions create a linear gradient in temperature across the domain, with three sides at 50 and one side at 100. The average temperature at the center, considering the influence of the boundaries, results in a value of 62.5.

Question 16

The “order” of the following ordinary differential equation is _____.

(d³ y)/(dx³) + ((d² y)/(dx²))⁶ + ((dy)/(dx))⁴ + y = 0

Accepted Answer

3. The order of a differential equation is determined by the highest derivative present in the equation. In this case, the highest derivative is the third derivative, ((d³ y)/(dx³), which indicates that the order of the equation is 3.

Question 17

Pick the CORRECT solution for the following differential equation

dy/dx = e^(x−y)

Accepted Answer

y = ln(e^x + Constant). The correct option is derived from separating variables and integrating both sides of the equation. By rearranging and integrating, we find that the solution takes the form of y expressed as the natural logarithm of a function involving e raised to x plus a constant.

Question 18

Consider the following recurrence relation: T(n) = 2T(n − 1) + n2ⁿ for n > 0, T(0) = 1. Which ONE of the following options is CORRECT?

Accepted Answer

T(n) = Θ(n² 2ⁿ). Let S(n)=T(n)/2^n. Dividing T(n)=2T(n-1)+n*2^n by 2^n gives S(n)=S(n-1)+n, so S(n)=Theta(n^2). Therefore T(n)=Theta(n^2 * 2^n), which is the first option, not Theta(4^n).

Question 19

Common Data for Questions 74 and 75:
Consider the following functions:

int f1 ( int n)
{
 if (n == 0 || n == 1)
 return n;
 else
 return (2*f1(n-1) + 3*f1(n-2));
}

int f2 ( int n)
{
 int i;
 int X[N], Y[N], Z[N];
 X[0] = Y[0] = Z[0] = 0;
 X[1] = 1; Y[1] = 2; Z[1] = 3;
 for (i = 2; i <= n

Accepted Answer

Θ(2ⁿ) and Θ(n). The function f1(n) exhibits exponential growth due to its recursive calls, leading to a time complexity of Θ(2ⁿ). In contrast, f2(n) uses an iterative approach with a linear loop, resulting in a time complexity of Θ(n).

Question 20

The Lucas sequence Lₙ is defined by the recurrence relation: Lₙ = Lₙ₋₁ + Lₙ₋₂, for n ≥ 3, with L₁ = 1 and L₂ = 3. Which one of the options given is TRUE?

Accepted Answer

Lₙ = ((1 + √5)/2)ⁿ + ((1 - √5)/2)ⁿ. For L1=1, L2=3 with Ln=Ln-1+Ln-2, the closed form is Ln = ((1+sqrt5)/2)^n + ((1-sqrt5)/2)^n, which gives L1=1 and L2=3. That is index 0, not the stored difference form (index 3) which would give a sqrt5 factor.

Question 21

Let T(n) be the recurrence relation defined as follows:

T(0) = 1,
T(1) = 2, and
T(n) = 5T(n − 1) − 6T(n − 2) for n ≥ 2

Which one of the following statements is TRUE?

Accepted Answer

T(n) = Θ(3ⁿ). The recurrence relation T(n) = 5T(n − 1) − 6T(n − 2) has a characteristic polynomial with roots that indicate exponential growth, specifically leading to a solution of the form T(n) = Θ(3ⁿ), which aligns with the dominant root of the characteristic equation.

Question 22

The solution of the differential equation k² d²y/dx² = y - y₂ under the boundary conditions (i) y = y₁ at x = 0 and (ii) y = y₂ at x = ∞, where k, y₁ and y₂ are constants, is

Accepted Answer

y = (y₁ - y₂) exp(-x/k) + y₂. The correct option accurately reflects the solution to the differential equation, incorporating the exponential decay factor that satisfies the boundary conditions, where the function approaches y₂ as x approaches infinity and equals y₁ at x = 0.

Question 23

The z-transform of a sequence x[n] is given by X[z] = 0.5/(1 - 2z⁻¹). It is given that the region of convergence of X[z] includes the unit circle. The value of x[0] is

Accepted Answer

0. X(z) = 0.5/(1 - 2 z^-1) has a pole at z = 2. Since the ROC must include the unit circle, the ROC is |z| < 2, making x[n] left-sided: x[n] = -0.5*2^n u[-n-1]. That sequence is zero for n >= 0, so x[0] = 0.

Question 24

Which of the following is a solution to the differential equation dx(t)/dt + 3x(t) = 0 ?

Accepted Answer

x(t) = 2e^(−3t). The correct option, x(t) = 2e^(−3t), satisfies the differential equation because it represents an exponential decay with a rate of 3, which matches the coefficient of x(t) in the equation. This solution can be verified by substituting it back into the equation and confirming that both sides are equal.

Question 25

A linear, time-invariant, causal continuous time system has a rational transfer function with simple poles at s = -2 and s = -4, and one simple zero at s = -1. A unit step u(t) is applied at the input of the system. At steady state, the output has constant value of 1. The impulse response

Accepted Answer

[-4 exp(-2t) + 12 exp(-4t)]u(t). With poles at -2,-4 and zero at -1, H(s)=K(s+1)/((s+2)(s+4)). Unit-step steady-state output 1 means H(0)=K/8=1, so K=8. Partial fractions give residues -4 at s=-2 and +12 at s=-4, so h(t)=[-4e^(-2t)+12e^(-4t)]u(t), which is option C, not the stored option D.

Question 26

Group I lists a set of four transfer functions. Group II gives a list of possible step responses y(t). Match the step responses with the corresponding transfer functions.

Group I
P = 25/(s² + 25)
Q = 36/(s² + 20s + 36)
R = 36/(s² + 12s + 36)
S = 49/(s² + 7s + 49)

Group II
1. [step respon

Accepted Answer

P-3, Q-2, R-4, S-1. The correct option matches each transfer function with its corresponding step response based on the characteristics of their poles and zeros. Transfer function P has a simple oscillatory response, Q exhibits a damped response, R shows a critically damped response, and S has an overdamped response, aligning perfectly with the given step response plots.

Question 27

A certain system has transfer function G(s) = (s + 8)/(s² + αs - 4), where α is a parameter. Consider the standard negative unity feedback configuration as shown below.

[block diagram of standard negative unity feedback with G(s)]

Which of the following statements is true?

Accepted Answer

For all positive values of α, the closed loop system is stable.. Unity feedback gives characteristic equation s^2+as-4 + (s+8) = s^2+(a+1)s+4 = 0. For a second-order polynomial all coefficients positive guarantees stability: a+1>0 and 4>0, so the system is stable for every positive a. The correct option is 'stable for all positive values of a'.

Question 28

The order of the differential equation d²y/dt² + (dy/dt)³ + y⁴ = e^(−t) is

Accepted Answer

2. The order of a differential equation is determined by the highest derivative present. In this case, the highest derivative is the second derivative, d²y/dt², which indicates that the order of the equation is 2.

Question 29

The ROC of Z-transform of the discrete time sequence x(n) = (1/3)ⁿ u(n) - (1/2)ⁿ u(n-1) is

Accepted Answer

|z| > 1/2. The correct option is right because the Z-transform converges for values of |z| greater than the largest pole of the sequence. In this case, the poles are at z = 1/3 and z = 1/2, so the region of convergence (ROC) is outside the largest pole, which is |z| > 1/2.

Question 30

Given that F(s) is the one-sided Laplace transform of f(t), the Laplace transform of ∫₀^t f(τ)dτ is

Accepted Answer

1/s F(s). The Laplace transform of the integral of a function is given by the property that states the transform of the integral from 0 to t of f(τ) is equal to 1/s times the Laplace transform of f(t), which is represented as 1/s F(s). This reflects the relationship between integration in the time domain and multiplication by 1/s in the frequency domain.

Question 31

Match each differential equation in Group I to its family of solution curves from Group II.

Group I
P. dy/dx = y/x
Q. dy/dx = -y/x
R. dy/dx = x/y
S. dy/dx = -x/y

Group II
1. Circles
2. Straight lines
3. Hyperbolas

Accepted Answer

P-2, Q-3, R-3, S-1. The correct option matches each differential equation to its corresponding family of solution curves based on their forms. For example, P represents a linear relationship leading to straight lines, while Q describes a negative relationship resulting in hyperbolas, and R and S also correspond to hyperbolas and circles respectively.

Question 32

A function n(x) satisfies the differential equation d² n(x)/dx² - n(x)/L² = 0 where L is a constant. The boundary conditions are: n(0)=K and n(∞)=0. The solution to this equation is

Accepted Answer

n(x) = K exp(-x/L). The correct option is right because it satisfies both the differential equation and the boundary conditions; specifically, the exponential decay form ensures that as x approaches infinity, n(x) approaches zero, while at x=0, it equals K.

Question 33

For a parallel RLC circuit, which one of the following statements is NOT correct?

Accepted Answer

At resonance, the magnitude of input impedance attains its minimum value. At resonance in a parallel RLC circuit, the impedance is at its maximum, not minimum, because the reactive components cancel each other out, leading to a purely resistive input impedance.

Question 34

Two discrete time systems with impulse responses h1[n] = δ[n − 1] and h2[n] = δ[n − 2] are connected in cascade. The overall impulse response of the cascaded system is

Accepted Answer

δ[n − 3]. The overall impulse response of two systems in cascade is found by convolving their individual impulse responses. Convolving h1[n] = δ[n − 1] with h2[n] = δ[n − 2] results in δ[n − 3], which represents the combined effect of the two systems.

Question 35

Consider a differential equation (dy(x))/(dx)-y(x)=x with the initial condition y(0)=0. Using Euler's first order method with a step size of 0.1, the value of y(0.3) is

Accepted Answer

0.031. With y'=y+x, y(0)=0, h=0.1: y(0.1)=0+0.1(0+0)=0; y(0.2)=0+0.1(0+0.1)=0.01; y(0.3)=0.01+0.1(0.01+0.2)=0.031. So y(0.3)=0.031, not 0.0631.

Question 36

Given f(t)=L⁻¹ [(3s+1)/(s³+4s²+(K-3)s) ]. If lim_(t→∞) f(t)=1, then the value of K is

Accepted Answer

4. By final value theorem, lim f(t)=lim_{s->0} sF(s)=(3s+1)/(s^2+4s+K-3) at s=0 = 1/(K-3)=1, so K-3=1 and K=4. Stored value 2 is wrong.

Question 37

In the circuit shown, the switch S is open for a long time and is closed at t=0. The current i(t) for t≥ 0⁺ is

Accepted Answer

i(t)=0.5-0.125e^(-1000t) A. The correct option reflects the behavior of the circuit after the switch is closed, where the steady-state current approaches 0.5 A and the transient response decays exponentially with a time constant determined by the circuit components, specifically matching the derived equation for the current.

Question 38

A continuous time LTI system is described by
(d² y(t))/dt² + 4(dy(t))/dt + 3y(t) = 2(dx(t))/dt + 4x(t)
Assuming zero initial conditions, the response y(t) of the above system for the input x(t) = e^(-2t)u(t) is given by

Accepted Answer

(e^(-t) - e^(-3t))u(t). The correct option is derived from the system's response to the given input using the Laplace transform method, where the poles of the system and the input's exponential form lead to a specific combination of terms in the output. The response reflects the system's behavior characterized by its differential equation, resulting in the specified output form.

Question 39

The solution of the differential equation dy/dx = ky, y(0) = c is

Accepted Answer

y = ce^(kx). The solution to the differential equation dy/dx = ky is derived from the separation of variables and integration, leading to an exponential function where the constant c represents the initial condition y(0) = c, resulting in y = ce^(kx).

Question 40

If x[n] = (1/3)^(|n|) - (1/2)ⁿ u[n], then the region of convergence (ROC) of its Z-transform in the z-plane will be

Accepted Answer

1/2 < |z| < 3. (1/3)^|n| is two-sided with ROC 1/3<|z|<3; (1/2)^n u[n] is right-sided with ROC |z|>1/2. The overall ROC is the intersection, 1/2<|z|<3, not 1/3<|z|.

Question 41

The impulse response of a system is h(t) = t u(t). For an input u(t-1), the output is

Accepted Answer

(t-1)²/2 u(t). The output of a linear time-invariant system can be found by convolving the input with the impulse response. In this case, the convolution of the input u(t-1) with the impulse response h(t) = t u(t) results in the output (t-1)²/2 u(t), which reflects the shifted nature of the input.

Question 42

A system is described by the differential equation d²y/dt² - 5 dy/dt = δ(t) * x(t).

Let x(t) be a rectangular pulse given by
x(t) = { 1, 0 ≤ t ≤ 2
 { 0, otherwise

Assuming that y(0) = 0 and dy/dt = 0 at t = 0, the Laplace transform of y(t) is

Accepted Answer

[1 - e^(-2s)] / [s(s-2)(s-3)]. The correct option represents the Laplace transform of the system's response to the input rectangular pulse, accounting for the initial conditions and the impulse response. The term [1 - e^(-2s)] captures the effect of the pulse from t=0 to t=2, while the denominator s(s-2)(s-3) reflects the system's dynamics as described by the differential equation.

Question 43

The Bode plot of a transfer function G(s) is shown in the figure below. The gain (20 log |G(s)|) is 32 dB and -8 dB at 1 rad/s and 10 rad/s respectively. The phase is negative for all ω. Then G(s) is

Accepted Answer

39.8 / s². The correct option is right because the transfer function G(s) = 39.8 / s² indicates a second-order system with a gain that decreases with frequency, which aligns with the given Bode plot characteristics of a gain of 32 dB at low frequencies and -8 dB at higher frequencies, along with a consistently negative phase.

Question 44

A discrete-time signal x[n] = sin(π² n), n being an integer, is

Accepted Answer

not periodic. The signal x[n] = sin(π² n) is not periodic because the frequency π² is an irrational multiple of π, which means that the signal does not repeat at regular intervals for integer values of n.

Question 45

A continuous, linear time-invariant filter has an impulse response h(t) described by
h(t) = { 3 for 0 ≤ t ≤ 3
 { 0 otherwise
When a constant input of value 5 is applied to this filter, the steady state output is _____.

Accepted Answer

45. The output of a linear time-invariant filter to a constant input can be calculated by multiplying the input value by the area under the impulse response. Here, the area under h(t) is 3 (height) multiplied by 3 (width), which equals 9, and when this is multiplied by the constant input of 5, the result is 45.

Question 46

Let x[n] = (-1/9)ⁿ u(n) - (-1/3)ⁿ u(-n-1). The Region of Convergence (ROC) of the z-transform of x[n]

Accepted Answer

is 1/3 > |z| > 1/9.. The ROC is determined by the behavior of the two components of the signal: the first term converges for |z| > 1/9 and the second term converges for |z| < 1/3. Therefore, the overall ROC is the intersection of these two regions, which results in the range 1/3 > |z| > 1/9.

Question 47

A system is described by the following differential equation, where u(t) is the input to the system and y(t) is the output of the system.

y'(t) + 5y(t) = u(t)

When y(0) = 1 and u(t) is a unit step function, y(t) is

Accepted Answer

0.2 + 0.8e⁻⁵t. The correct option is derived from solving the first-order linear differential equation using the method of integrating factors. The particular solution for a unit step input leads to a steady-state output of 0.2, while the homogeneous solution decays over time, resulting in the final expression of 0.2 + 0.8e⁻⁵t.

Question 48

If the characteristic equation of the differential equation d²y/dx² + 2α dy/dx + y = 0 has two equal roots, then the values of α are

Accepted Answer

±1. Characteristic equation is r^2 + 2(alpha)r + 1 = 0. Equal roots require discriminant (2 alpha)^2 - 4 = 0, so alpha^2 = 1 and alpha = +/-1. Correct option is +/-1, not +/-1/2.

Question 49

The input-output relationship of a causal stable LTI system is given as y[n] = α y[n−1] + β x[n]. If the impulse response h[n] of this system satisfies the condition ∑n=0∞ h[n] = 2, the relationship between α and β is

Accepted Answer

α = 1 − β/2. The condition ∑n=0∞ h[n] = 2 indicates that the system is stable and the sum of the impulse response is finite. For a causal stable LTI system, the relationship between the coefficients α and β must satisfy the equation derived from the system's response, leading to α = 1 - β/2 as the correct relationship.

Question 50

Which ONE of the following is a linear non-homogeneous differential equation, where x and y are the independent and dependent variables respectively?

Accepted Answer

dy/dx + xy = e^-x. This equation is linear because it can be expressed in the form of a linear combination of the dependent variable and its derivatives, and it is non-homogeneous due to the presence of the term e^-x on the right side, which is not a function of y.

GATE Engineering Mathematics: Differential Equations questions with solutions

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