1Which of the following defines the order of a partial differential equation (PDE)?
A.The power of the highest order derivative
B.The order of the highest order derivative occurring in the equation
C.The number of independent variables
D.The number of dependent variables
Correct Answer: The order of the highest order derivative occurring in the equation
Explanation:The order of a PDE is defined as the order of the highest partial derivative appearing in the equation.
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2In the standard notation for PDEs involving a dependent variable and independent variables and , what does the symbol represent?
A.
B.
C.
D.
Correct Answer:
Explanation:Standard notations are , , , , and .
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3A partial differential equation is said to be linear if:
A.The dependent variable and its derivatives appear with power 1 and are not multiplied together
B.The dependent variable appears with power 2
C.The partial derivatives are multiplied together
D.The independent variables appear with power 1
Correct Answer: The dependent variable and its derivatives appear with power 1 and are not multiplied together
Explanation:Linearity requires that the dependent variable and all its partial derivatives occur linearly (degree 1) and no products of the dependent variable and/or its derivatives exist.
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4Consider the general second-order linear PDE: . The equation is classified as hyperbolic if:
A.
B.
C.
D.
Correct Answer:
Explanation:The discriminant determines the classification. If it is positive (), the PDE is hyperbolic.
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5The one-dimensional wave equation is classified as which type of PDE?
A.Parabolic
B.Elliptic
C.Hyperbolic
D.Linear homogeneous ODE
Correct Answer: Hyperbolic
Explanation:Rewriting as , we identify , , (if is ). . Since , the discriminant is positive, making it Hyperbolic.
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6The one-dimensional heat equation is classified as:
A.Elliptic
B.Parabolic
C.Hyperbolic
D.Circular
Correct Answer: Parabolic
Explanation:Here (coefficient of ), , and (coefficient of is 0). . When the discriminant is zero, the equation is parabolic.
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7The two-dimensional Laplace equation is classified as:
A.Elliptic
B.Parabolic
C.Hyperbolic
D.Linear Non-homogeneous
Correct Answer: Elliptic
Explanation:Here , , . . A negative discriminant indicates an elliptic PDE.
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8What is the degree of the PDE ?
A.1
B.2
C.3
D.6
Correct Answer: 2
Explanation:The degree is the power of the highest order derivative appearing in the equation. The highest order is 3, and its power is 2.
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9Which of the following equations represents the standard one-dimensional Wave Equation?
A.
B.
C.
D.
Correct Answer:
Explanation:The wave equation involves the second derivative with respect to time () proportional to the second derivative with respect to space ().
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10In the method of separation of variables for , the solution is assumed to be of the form:
A.
B.
C.
D.
Correct Answer:
Explanation:The method assumes the solution can be written as a product of functions, each depending on a single independent variable.
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11When applying the method of separation of variables to with , the resulting ODE for is (where is the separation constant):
A.
B.
C.
D.
Correct Answer:
Explanation:Substituting gives . Dividing by gives . Equating to a constant , we get or .
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12In the solution of the Wave Equation using separation of variables, the separation constant is usually chosen as (a negative constant) to ensure:
A.Exponentially increasing solutions
B.Logarithmic solutions
C.Periodic (oscillatory) solutions
D.Linear solutions
Correct Answer: Periodic (oscillatory) solutions
Explanation:For physical wave problems (vibrating strings), the solution must be periodic in time and space. A negative separation constant leads to trigonometric solutions (sine/cosine).
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13Which of the following is the most general solution to ?
A.
B.
C.
D.
Correct Answer:
Explanation:The characteristic equation is , so . This results in a trigonometric general solution.
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14In the one-dimensional wave equation describing a vibrating string, what does represent?
A.
B.
C.
D.
Correct Answer:
Explanation:For a vibrating string, , where is the tension and is the linear mass density.
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15What are the boundary conditions for a string of length fixed at both ends ( and )?
A. and for all
B. and for all
C. and
D. and
Correct Answer: and for all
Explanation:Fixed ends imply that the displacement is zero at the endpoints for all time .
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16For a string fixed at both ends, the possible values of the eigenvalue are:
A. where
B. where
C.
D.Any real number
Correct Answer: where
Explanation:Applying boundary conditions and to yields , so , thus .
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17The general solution of the wave equation for a string of length fixed at both ends is given by:
A.
B.
C.
D.
Correct Answer:
Explanation:This represents the superposition of standing waves (normal modes) vibrating at different frequencies.
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18If a vibrating string is released from rest, which initial condition applies?
A.
B.
C.
D.
Correct Answer:
Explanation:Released from rest means the initial velocity is zero.
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19For the wave equation solution with zero initial velocity, the coefficient associated with the term is:
A.
B.1
C.
D.Undefined
Correct Answer:
Explanation:Taking the time derivative of the general solution, the sine term becomes cosine and the cosine term becomes sine. At , the sine term vanishes. For velocity to be 0, the coefficients of the resulting cosine terms (derived from the original sine terms, ) must be zero.
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20The fundamental period of vibration for a string of length is:
A.
B.
C.
D.
Correct Answer:
Explanation:The fundamental frequency is . The period is .
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21In the Heat Equation , the constant is known as:
A.Thermal conductivity
B.Specific heat
C.Thermal diffusivity
D.Young's modulus
Correct Answer: Thermal diffusivity
Explanation: is the thermal diffusivity, where is conductivity, is specific heat, and is density.
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22The condition for steady-state temperature distribution is:
A.
B.
C.
D.
Correct Answer:
Explanation:Steady state means the temperature distribution does not change with time.
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23The steady-state solution of the one-dimensional heat equation is:
A.
B.
C.
D.
Correct Answer:
Explanation:Integrating twice with respect to yields a linear function .
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24Which of the following describes the transient part of the solution to the heat equation?
A.It increases exponentially with time.
B.It remains constant with time.
C.It decays exponentially to zero as .
D.It oscillates indefinitely.
Correct Answer: It decays exponentially to zero as .
Explanation:The time-dependent part of the heat equation solution involves terms like , which decay to zero as time progresses.
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25For a rod of length with ends kept at temperatures $0$ (i.e., ), the solution involves terms of the form:
A.
B.
C.
D.
Correct Answer:
Explanation:The boundary conditions and select the sine function in space, and the heat equation dictates exponential decay in time.
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26If the ends of a rod are thermally insulated, the boundary conditions are:
A.
B.
C.
D.
Correct Answer:
Explanation:Insulation implies no heat flow across the boundaries. Heat flow is proportional to the temperature gradient . Thus, the gradient must be zero.
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27Which constant is used in separating variables for the Heat Equation to get a bounded physical solution?
A.
B.
C.
D.
Correct Answer:
Explanation:A negative separation constant ensures that the time component decays exponentially (), consistent with physical cooling/diffusion.
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28As , the temperature of a rod with insulated ends approaches:
A.
B.The initial temperature at the center
C.The average of the initial temperature distribution
D.Infinity
Correct Answer: The average of the initial temperature distribution
Explanation:In an insulated rod, heat energy is conserved. Eventually, the temperature equalizes to the mean value of the initial profile.
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29The Laplace equation in two dimensions is explicitly written as:
A.
B.
C.
D.
Correct Answer:
Explanation:This is the standard Cartesian form of the 2D Laplace equation.
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30Solutions to the Laplace equation are called:
A.Harmonic functions
B.Periodic functions
C.Transient functions
D.Heaviside functions
Correct Answer: Harmonic functions
Explanation:Any function that satisfies Laplace's equation is defined as a harmonic function.
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31In solving Laplace's equation for a rectangular plate, if the boundary conditions are zero on the vertical edges (), the solution typically involves:
A. and hyperbolic functions of
B. and hyperbolic functions of
C.Exponential functions of and
D.Polynomials of and
Correct Answer: and hyperbolic functions of
Explanation:Zero boundary conditions on suggest trigonometric solutions in (eigenfunctions), which forces the component to be hyperbolic (exponential-like) to satisfy the elliptic PDE.
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32The two-dimensional steady-state heat flow is governed by:
A.Wave Equation
B.Heat Equation
C.Laplace Equation
D.Poisson Equation
Correct Answer: Laplace Equation
Explanation:Steady state implies time independence (). The heat equation reduces to .
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33Which of the following is a possible solution set for using separation of variables ()?
A.
B.
C.
D.
Correct Answer:
Explanation:For Laplace, if one variable has trigonometric solutions (negative separation constant), the other must have hyperbolic solutions (positive separation constant) so that their second derivatives sum to zero.
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34A problem requiring the solution of Laplace's equation inside a region with values of the function prescribed on the boundary is called a:
A.Neumann problem
B.Dirichlet problem
C.Cauchy problem
D.Robin problem
Correct Answer: Dirichlet problem
Explanation:The Dirichlet problem specifies the values of the function itself along the boundary.
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35A problem requiring the solution of Laplace's equation where the normal derivative is specified on the boundary is called a:
A.Neumann problem
B.Dirichlet problem
C.Initial value problem
D.Mixed problem
Correct Answer: Neumann problem
Explanation:The Neumann problem specifies the values of the derivative (gradient/flux) on the boundary.
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36What is the form of Laplace's equation in polar coordinates ?
A.
B.
C.
D.
Correct Answer:
Explanation:This is the standard transformation of the Laplacian operator to polar coordinates.
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37In the solution of Laplace's equation in polar coordinates for a circular disk, the solution must be periodic in with period:
A.
B.
C.
D.Infinite
Correct Answer:
Explanation:For the solution to be single-valued and physically meaningful on a full circle, .
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38Which principle states that a harmonic function defined on a bounded domain takes its maximum and minimum values on the boundary?
A.Superposition Principle
B.Maximum Modulus Principle
C.Maximum Principle
D.D'Alembert's Principle
Correct Answer: Maximum Principle
Explanation:The Maximum Principle is a core property of harmonic functions (solutions to Laplace equations), stating extrema occur at the boundaries.
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39When solving the wave equation, if the string has length , the term represents:
A.The time evolution factor
B.The normal modes of vibration (spatial shape)
C.The damping factor
D.The initial velocity
Correct Answer: The normal modes of vibration (spatial shape)
Explanation:The spatial sine terms represent the standing wave patterns or normal modes.
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40What is the value of for the heat equation if the initial temperature is zero everywhere and the boundary temperatures are zero?
A.
B.
C.
D.
Correct Answer:
Explanation:With no initial heat energy and no heat source at the boundaries, the temperature remains zero.
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41Which of the following PDEs represents the vibration of a string?
A.Laplace Equation
B.Poisson Equation
C.Heat Equation
D.Wave Equation
Correct Answer: Wave Equation
Explanation:The Wave Equation models propagation of waves, such as sound, light, and vibrating strings.
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42Using separation of variables on , if the separation constant is , what is the solution for ?
A.
B.
C.
D.
Correct Answer:
Explanation:The separated equation for is . Here is the separation constant (often denoted ). Wait, checking convention. Usually . If the constant itself is given as -16, then , so . Note: In some conventions . If the constant given applies directly to the ratio, .
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43In the D'Alembert solution of the wave equation , the term represents:
A.A wave traveling in the positive x-direction
B.A wave traveling in the negative x-direction
C.A standing wave
D.A decaying wave
Correct Answer: A wave traveling in the negative x-direction
Explanation:Functions of represent waves moving to the left (negative direction), while move to the right.
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44For the heat equation with boundary conditions and , the steady state solution is:
A.
B.
C.
D.
Correct Answer:
Explanation:This is the linear interpolation between the two boundary temperatures over the length .
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45The classification of is:
A.Elliptic
B.Parabolic
C.Hyperbolic
D.None of these
Correct Answer: Parabolic
Explanation:. Discriminant . Hence Parabolic.
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46Which superposition of terms solves the 2D Laplace equation on a rectangle with on all boundaries except ?
A.
B.
C.
D.
Correct Answer:
Explanation:The term in satisfies . The term in satisfies (since ) and allows for a non-zero value at .
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47A string of length is plucked at the center . This excites:
A.Only even harmonics
B.Only odd harmonics
C.All harmonics
D.No harmonics
Correct Answer: Only odd harmonics
Explanation:Plucking at the center creates a symmetric shape. Even harmonics (like ) have a node at the center (are antisymmetric around the center or zero there), so they are not excited. Only odd harmonics are present.
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48What is the physical interpretation of the boundary condition in the context of a vibrating string?
A.Fixed end
B.Free end (sliding ring without friction)
C.Damped end
D.Forced vibration
Correct Answer: Free end (sliding ring without friction)
Explanation:A zero slope at the end of a string implies there is no transverse force component acting to pull it back, characteristic of a free end.
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49The method of separation of variables converts a PDE of independent variables into: