Chapter 22: Differential Equations (PDE) and Total Differential Equations (Set-5)
For the system x′ = 3x + 4y, y′ = −4x + 3y, the eigenvalues of A are
A 3 ± 4i
B 4 ± 3i
C 7, −1
D 1, −7
The matrix is (34−43)(3−443). Its trace is 6 and determinant is 32+42=2532+42=25. Characteristic equation gives λ=3±4iλ=3±4i, implying spiral-type solutions with growth e3te3t.
For x′ = y, y′ = x, eliminating y gives the single equation
A x″ + x = 0
B x′ − x = 0
C x″ − x = 0
D x′ + x = 0
Differentiate x′=y to get x″=y′. Using y′=x gives x″=x, or x″−x=0. Solutions are x=c1et+c2e−tx=c1et+c2e−t, then y=x′.
For a 2×2 linear system with A having eigenvalues 2 and −2, the phase portrait is generally
A Spiral source
B Spiral sink
C Center
D Saddle point
Real eigenvalues with opposite signs indicate one stable and one unstable direction. Trajectories approach along the negative eigenvalue eigenvector and diverge along the positive eigenvalue direction.
If a constant-coefficient system has complex eigenvalues with negative real part, solutions approach
A Spiral away origin
B Spiral into origin
C Straight line only
D Constant circle
Complex eigenvalues a±bia±bi produce oscillations with factor eateat. If a<0a<0, the exponential decays, so trajectories spiral inward toward the equilibrium point.
The system dx/P = dy/Q = dz/R is most directly associated with
A Laplace PDE
B Heat equation
C Lagrange linear PDE
D Bernoulli ODE
For Lagrange PDE Pp+Qq=RPp+Qq=R with p=∂z/∂xp=∂z/∂x, q=∂z/∂yq=∂z/∂y, the characteristic (auxiliary) equations are dx/P=dy/Q=dz/Rdx/P=dy/Q=dz/R, used to find two first integrals.
In Lagrange PDE, if two independent integrals are u and v, the solution is
A u+v=0 only
B u=v only
C z=u+v
D F(u,v)=0
Two independent first integrals remain constant along characteristics. Any functional relation between them, F(u,v)=0F(u,v)=0, generates a family of solution surfaces satisfying the PDE.
If a first-order PDE has form F(x,y,z,p,q)=0, then Charpit’s method is intended for
A Linear second-order PDE
B Exact total differential
C Nonlinear first-order PDE
D Linear simultaneous ODE
Charpit’s method constructs characteristic-type equations for nonlinear first-order PDEs involving p and q. It aims to obtain a complete integral containing parameters, then derive particular solutions.
For a PDE, “degree” is not defined when derivatives appear
A In polynomial form
B Inside trigonometric
C As linear terms
D As constant terms
Degree requires the PDE to be polynomial in derivatives. If derivatives appear inside functions like sin(ux) or exp(ux), the PDE is not polynomial in derivatives, so degree is undefined.
Consider Mdx+Ndy=0 with M=2xy and N=x². The equation is
A Not exact
B Homogeneous PDE
C Hyperbolic PDE
D Exact
Compute ∂M/∂y=2x and ∂N/∂x=2x. Since they are equal, the form is exact. A potential exists, so integrating gives an implicit solution F(x,y)=C.
For the same M and N, one potential function can be
A xy²
B x²+y²
C x²y
D x+y
Integrate M=2xy w.r.t. x: F=x2y+g(y)F=x2y+g(y). Then Fy=x²+g′(y) must equal N=x², giving g′(y)=0. Thus F=x²y is a valid potential.
If Mdx+Ndy=0 is not exact, an integrating factor μ(x) can exist when
A (My−Nx)/N is x-only
B (Mx−Ny)/M is x-only
C (My+Nx)/N is y-only
D (Mx+Ny)/M is y-only
A standard criterion for μ(x) is that (∂M/∂y−∂N/∂x)/N(∂M/∂y−∂N/∂x)/N depends only on x. Then μ(x)=exp(∫ … dx) makes the equation exact.
For μ(y) depending only on y, a standard test uses
A (My−Nx)/N is x-only
B (Mx−Ny)/N is y-only
C (Nx+My)/M is x-only
D (Nx−My)/M is y-only
If (∂N/∂x−∂M/∂y)/M(∂N/∂x−∂M/∂y)/M depends only on y, an integrating factor μ(y) exists. Then μ(y)=exp(∫ … dy) converts the form to exact.
In second-order PDE A uxx + 2B uxy + C uyy, the discriminant used is
A B²+AC
B A²−BC
C B²−AC
D C²−AB
The discriminant is B²−AC for the standard Auxx+2Buxy+CuyyAuxx+2Buxy+Cuyy form. Its sign determines the PDE type: hyperbolic, parabolic, or elliptic.
The PDE is parabolic when
A B²−AC = 0
B B²−AC < 0
C B²−AC > 0
D A = 0 only
Zero discriminant indicates repeated characteristic direction. This is the defining feature of parabolic PDEs, which behave like diffusion equations such as the heat equation.
The PDE is hyperbolic when
A B²−AC = 0
B B²−AC < 0
C B²−AC > 0
D B = 0 only
Positive discriminant gives two distinct real characteristic families. This corresponds to wave-like propagation and is the hallmark of hyperbolic equations such as the wave equation.
The PDE is elliptic when
A B²−AC > 0
B B²−AC < 0
C B²−AC = 0
D C = 0 only
Negative discriminant implies no real characteristics. This is typical of equilibrium field problems like Laplace’s equation, where boundary values determine the solution inside the region.
For Laplace equation uxx + uyy = 0, the type is
A Hyperbolic
B Parabolic
C Nonlinear
D Elliptic
Laplace’s equation has A=1, B=0, C=1 so B²−AC=−1<0. Therefore it is elliptic, typically solved as a boundary value problem with strong uniqueness properties.
For the heat equation ut = k uxx, the type (in x,t variables) is
A Elliptic
B Hyperbolic
C Parabolic
D Exact
Heat equation is first order in time and second order in space, matching diffusion behavior. In standard classification of second-order PDEs, it corresponds to parabolic type with repeated characteristics.
For the wave equation utt = c² uxx, the type is
A Elliptic
B Hyperbolic
C Parabolic
D Pfaffian
The wave equation has two real characteristic lines, representing propagation at finite speed. This is the defining behavior of hyperbolic PDEs and leads to solutions determined by initial data.
For a Neumann problem on Laplace’s equation, uniqueness is generally up to
A Multiplicative constant
B Sine function
C Quadratic term
D Additive constant
If only normal derivatives are specified on the boundary, adding a constant does not change derivatives. Thus solutions are not unique unless one extra condition fixes the constant level.
For a Dirichlet problem on Laplace’s equation, specifying u on boundary typically gives
A Unique solution
B Two solutions always
C No solution ever
D Infinite constants
For reasonable domains and boundary data, Dirichlet conditions usually ensure uniqueness for Laplace’s equation. This is a core result behind many physical steady-state potential problems.
In separation of variables, boundary conditions lead to an eigenvalue problem because
A PDE becomes nonlinear
B Exactness is forced
C Nontrivial solutions needed
D Degree becomes undefined
After separating u=X(x)T(t), the spatial ODE must satisfy boundary conditions. Nontrivial X exists only for special eigenvalues, producing discrete modes used in Fourier series solutions.
For fixed ends in a string, the spatial eigenfunctions are typically
A Cosine series
B Exponential series
C Polynomial series
D Sine series
Fixed ends require displacement zero at both ends. Sine functions satisfy sin(0)=0 and sin(nπ)=0 naturally, making them the standard eigenfunctions in wave and heat problems with fixed boundaries.
If a rod end is insulated, the boundary condition is best expressed as
A u = 0
B ux = 0
C ut = 0
D uxx = 0
Insulation means no heat flux across the boundary. Since flux is proportional to the spatial derivative, the normal derivative ux is set to zero, which is a Neumann boundary condition.
In well-posedness, “existence” means
A Exactly one solution
B Solution is stable
C At least one solution
D Solution is periodic
Existence requires that a solution actually can be found for the given data. Uniqueness and stability are separate requirements, together ensuring the problem is mathematically meaningful.
In well-posedness, “uniqueness” means
A Only one solution
B No solution exists
C Infinite solutions always
D Solution is constant
Uniqueness ensures the same input data cannot produce multiple different solutions. This is essential for predicting physical behavior and ensuring computations represent a single correct outcome.
In well-posedness, “stability” means
A Data irrelevant to solution
B Only boundary matters
C Only time matters
D Continuous dependence on data
Stability means small changes in initial or boundary data cause only small changes in the solution. Without stability, measurement errors could create huge prediction errors, making the model unreliable.
In a linear PDE, superposition fails only if
A PDE is elliptic
B PDE is parabolic
C PDE is nonlinear
D PDE is hyperbolic
Superposition is a property of linear operators. Elliptic, parabolic, and hyperbolic are types of PDEs; any of them can be linear. Nonlinearity is what breaks superposition.
A nonlinear first-order PDE may include
A Pp+Qq term
B p·q term
C p+q term
D p−q term
Products like p·q or p² create nonlinearity in derivatives. Lagrange’s PDE Pp+Qq=R is linear in p and q. Nonlinear terms require methods like Charpit’s.
In Lagrange PDE, characteristics are curves along which
A discriminant becomes zero
B degree becomes one
C boundary vanishes
D z changes consistently
Along characteristic curves, x, y, and z satisfy linked ODEs from dx/P=dy/Q=dz/R. This makes the PDE integrable along these curves and helps construct the full solution surface.
If one first integral is u(x,y,z)=c1, then along each characteristic
A u becomes linear
B u becomes zero
C u remains constant
D u becomes periodic
A first integral is constant by definition on characteristics. It labels characteristic curves/surfaces, and together with a second independent integral it builds the general solution relation.
In total differentials, path independence is equivalent to
A Exact differential form
B Hyperbolic PDE type
C Neumann boundary data
D Fourier convergence only
If a differential form equals dF, its integral between two points depends only on endpoints. This path independence is a key property of exact differentials and potential functions.
In a 3-variable form Mdx+Ndy+Pdz, “potential function” means
A Only boundary function
B Only time function
C Only eigenfunction
D F with dF form
A potential function F(x,y,z) exists if Mdx+Ndy+Pdz equals dF. Then the differential equation becomes dF=0, producing the implicit solution F = constant.
A “Pfaffian equation” is best described as
A Second-order PDE only
B Two-form equals zero
C One-form equals zero
D Determinant equals zero
A Pfaffian equation is a first-degree differential form set to zero, like Mdx+Ndy+Pdz=0. Solving involves integrability conditions or integrating factors to obtain a potential relation.
In a coupled system, if trace(A)=0 and det(A)>0 with real entries, the eigenvalues are
A Pure imaginary pair
B Real opposite signs
C Repeated positive
D All zeros
Characteristic equation is λ²−(trace)λ+det=0. With trace=0, λ²+det=0 gives λ=±i√det. This produces center-type oscillations in the linear system.
For a 2×2 linear system, if det(A)<0, then eigenvalues must be
A Both positive real
B Both negative real
C Opposite signs real
D Pure imaginary
For real 2×2 matrices, det equals product of eigenvalues. If det<0, the product is negative, so eigenvalues must have opposite signs, giving a saddle-type equilibrium.
If a boundary condition is u(0,t)=0 and u(L,t)=0, it is
A Nonhomogeneous Dirichlet
B Homogeneous Neumann
C Mixed Neumann
D Homogeneous Dirichlet
Dirichlet conditions specify the function value on boundaries. Since both values are zero, they are homogeneous Dirichlet conditions, commonly used for fixed-end strings or zero-temperature ends.
If boundary condition is ux(0,t)=0 and ux(L,t)=0, it is
A Homogeneous Dirichlet
B Homogeneous Neumann
C Nonhomogeneous Neumann
D Mixed Dirichlet
Neumann conditions specify derivative (flux). Setting derivatives to zero at boundaries means no flux and gives homogeneous Neumann conditions, typical for insulated ends in heat flow.
The “method of characteristics” is most directly used for
A Only Laplace equation
B Only Fourier expansions
C First-order PDE solving
D Only eigenvalue problems
Characteristics convert first-order PDEs into ODEs along curves. Solving those ODEs provides integral relations used to construct the solution surface, making it a central method for first-order PDEs.
If the PDE is linear and homogeneous, and u is a solution, then ku is
A Never a solution
B Only sometimes
C Also a solution
D Only if k=1
Homogeneous linear PDEs satisfy scaling: multiplying a solution by a constant keeps the equation balanced because each term scales equally. This is part of the superposition principle.
In Fourier method, orthogonality is mainly used to
A Compute coefficients
B Change PDE type
C Find discriminant
D Make PDE exact
Orthogonality allows isolating each mode coefficient by integrating against the corresponding eigenfunction. This makes coefficient calculation systematic and ensures expansions match initial/boundary data.
For wave equation on (0,L) with fixed ends, allowed eigenvalues are typically
A nπ/L values
B any real number
C only zero value
D only imaginary
Fixed-end conditions X(0)=X(L)=0 lead to eigenfunctions sin(nπx/L). This requires eigenvalues proportional to nπ/L, giving discrete standing wave modes.
For heat equation with zero-end temperatures, time factor usually decays like
A e^(+kλ² t)
B cos(λt) only
C sin(λt) only
D e^(−kλ² t)
Separation gives T′/T=−kλ², so T(t)=e^(−kλ² t). Negative exponent reflects diffusion damping: higher-frequency spatial modes decay faster, smoothing temperature over time.
For wave equation separation, time factor usually involves
A only exponential decay
B only polynomial growth
C sin and cos
D only constant term
Separation gives T″ + c²λ² T=0, whose solutions are sinusoidal. This reflects oscillatory behavior of waves, with frequency determined by λ and wave speed c.
In Charpit’s method, the “complete integral” typically contains
A One parameter
B Two parameters
C No parameter
D Only boundary data
For a first-order PDE in two variables, a complete integral generally contains two arbitrary constants. Charpit’s method aims to produce such a complete integral for nonlinear PDEs.
A first-order PDE solution “surface” is usually determined once
A discriminant is negative
B boundary is Neumann
C Initial curve is given
D equation is elliptic
General first-order PDE solutions contain an arbitrary function. Specifying data on an initial curve provides the needed condition to fix that function and select a unique solution surface.
In PDE classification, “characteristics” for hyperbolic equations represent
A Flux always zero
B Exactness checks
C Eigenvector directions
D Signal travel paths
For hyperbolic PDEs, characteristics show directions along which disturbances propagate. They help determine how initial data influences the solution and are fundamental in wave motion analysis.
A boundary condition plus initial condition set is typical for
A Parabolic PDE problems
B Elliptic PDE problems
C Pfaffian equations
D Exact ODE only
Parabolic equations like the heat equation evolve in time, so they need initial conditions. Because the domain is finite, boundary conditions are also required to ensure uniqueness and physical meaning.
If derivatives appear inside an exponential like e^(ux), then the PDE degree is
A Exactly one
B Exactly two
C Not defined
D Always zero
Degree is defined only when the PDE is polynomial in derivatives. Expressions like e^(ux) are non-polynomial in derivatives, so the degree cannot be assigned in the standard sense.
In total differential equations, after finding F, the final solution is written as
A F = 0 only
B dF = 1
C ∂F = 0
D F = constant
If Mdx+Ndy+Pdz is exact, it equals dF. The differential equation becomes dF=0, whose integral is F(x,y,z)=C. This is the standard implicit solution form.