Chapter 22: Differential Equations (PDE) and Total Differential Equations (Set-4)
For a linear system dX/dt = AX, if A has two distinct real eigenvalues, the solution is generally
A Pure polynomial only
B One constant mode
C Pure sine only
D Two exponential modes
Distinct real eigenvalues produce two independent eigenvectors, leading to two exponential terms. The general solution is a linear combination of these modes, with constants fixed by initial conditions.
If a 2×2 matrix A has a repeated eigenvalue but only one eigenvector, the system solution involves
A Only constants
B Only sine waves
C t·e^(λt) term
D Only e^(λt)
A defective matrix (insufficient eigenvectors) produces a generalized eigenvector. This adds a polynomial factor t multiplying e^(λt), giving the typical t·e^(λt) behavior in one solution component.
In elimination for coupled ODEs, a common sign you did it right is
A One variable removed
B PDE order increases
C Boundary terms appear
D Degree becomes undefined
Elimination aims to form a single differential equation in one unknown. If the eliminated variable still appears, elimination is incomplete, and the reduction step must be corrected.
For a system dx/dt = y and dy/dt = −x, the combined single equation for x is
A x′ − x = 0
B x″ + x = 0
C x′ + x = 0
D x″ − x = 0
Differentiate dx/dt=y to get x″=dy/dt. Substitute dy/dt=−x to obtain x″=−x, or x″+x=0. This is simple harmonic motion with sinusoidal solutions.
In a homogeneous linear system, the trivial solution always exists because
A Degree requires it
B Boundary forces it
C Zero satisfies equations
D Fourier gives it
In homogeneous systems, substituting all variables as zero makes every term zero, satisfying each equation. Nontrivial solutions depend on the system structure and initial conditions.
For a 2-variable exact differential equation Mdx+Ndy=0, the potential function F satisfies
A Fx = M and Fy = N
B Fxx = M only
C Fyy = N only
D Fx = N and Fy = M
If Mdx+Ndy equals dF, then dF=Fx dx + Fy dy. Matching coefficients gives Fx=M and Fy=N. Integrating M w.r.t. x and adjusting using Fy gives F.
If ∂M/∂y ≠ ∂N/∂x, then Mdx+Ndy=0 is
A Always separable
B Always exact
C Not exact
D Always linear PDE
Exactness requires equality of mixed partial derivatives from a potential function. If ∂M/∂y differs from ∂N/∂x, no potential function F exists directly, so an integrating factor may be needed.
A common integrating factor for Mdx+Ndy=0 depends on x only if
A (∂M/∂y−∂N/∂x)/N is x-only
B M/N is constant
C M and N equal
D (∂M/∂y+∂N/∂x)/M is y-only
If an integrating factor μ(x) exists, a standard test checks whether (∂M/∂y−∂N/∂x)/N is purely a function of x. Then μ(x)=exp(∫ … dx) makes the equation exact.
Similarly, an integrating factor depending on y only is possible if
A M is constant
B (∂N/∂x−∂M/∂y)/M is y-only
C M+N is zero
D (∂M/∂y−∂N/∂x)/N is x-only
For μ(y), a standard criterion is that (∂N/∂x−∂M/∂y)/M depends only on y. Then μ(y)=exp(∫ … dy) converts the differential form into an exact one.
A Pfaffian differential equation is most closely associated with
A Only dy/dx form
B Only p+q form
C General form Mdx+Ndy+Pdz
D Only uxx+uyy form
Pfaffian form refers to a general first-degree differential expression in several variables. Solving often involves integrability conditions or integrating factors to obtain a potential function.
In Lagrange’s PDE, one first integral can be obtained by choosing multipliers l,m,n such that
A lP+mQ+nR = 0
B lP+mQ+nR = z
C lP+mQ+nR = x
D lP+mQ+nR = 1
With dx/P=dy/Q=dz/R, if lP+mQ+nR=0 then ldx+mdy+ndz=0 along characteristics. If this differential is integrable, it produces a first integral u=c1.
In Lagrange PDE, after obtaining two independent integrals 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 integrals remain constant along characteristics. Any relation F(u,v)=0 defines a family of surfaces satisfying the PDE, giving the general integral of the Lagrange equation.
A “particular integral” differs from the “general integral” mainly because it
A Removes derivatives
B Changes PDE order
C Uses a condition
D Changes PDE type
The general integral contains an arbitrary function. A particular integral is chosen by applying an initial curve or constraint, which removes the arbitrariness and fixes the solution uniquely.
Charpit’s method deals with nonlinear first-order PDE F(x,y,z,p,q)=0 by forming characteristic-type equations involving p and q. Solving them leads to a complete integral.
In second-order PDE classification for A uxx + 2B uxy + C uyy, hyperbolic means
A B² − AC > 0
B B² − AC < 0
C B² − AC = 0
D A = 0 always
The discriminant B²−AC decides the type. Positive indicates two distinct real characteristic directions, which is the key feature of hyperbolic equations like the wave equation.
For the same form, parabolic means
A B² − AC > 0
B B² − AC < 0
C B² − AC = 0
D C = 0 always
Zero discriminant indicates a repeated characteristic direction. This is typical of diffusion-style equations, like heat flow, and is the defining feature of parabolic PDEs.
For the same form, elliptic means
A B² − AC < 0
B B² − AC = 0
C B² − AC > 0
D B = 0 always
Negative discriminant means no real characteristic directions in the usual sense. This matches elliptic PDEs like Laplace’s equation, often solved as boundary value problems.
Canonical variables are introduced mainly to simplify the
A Boundary data only
B Initial constants only
C Second-derivative part
D Fourier coefficients only
Changing variables can remove the mixed derivative term and reduce the second-order part to a standard canonical form. This makes classification clearer and helps apply suitable solution methods.
For elliptic equations like Laplace’s equation, uniqueness is typically tied to
A Initial velocity only
B Boundary conditions
C Discriminant only
D Integrating factor
Elliptic PDEs model steady states, so specifying boundary values on the domain usually determines a unique solution. This is why boundary value problems are central to elliptic PDE theory.
For wave equations, data is commonly specified as
A Boundary value only
B Potential function only
C Integrating factor only
D Initial displacement, velocity
The wave equation is second order in time, so two initial conditions are needed: displacement and velocity at the starting time. Boundary conditions may also be added for finite domains.
In separation of variables, boundary conditions help decide
A Integrating factor form
B PDE type always
C Eigenvalues allowed
D Exactness condition
Separation leads to eigenvalue problems in space. Boundary conditions restrict which eigenvalues and eigenfunctions are valid, determining the permitted modes in the final series solution.
In Fourier series solutions, coefficients are mainly found by matching
A Initial or boundary data
B Exactness condition
C Mixed derivative removal
D Discriminant sign
After writing the solution as a series, coefficients are chosen so the series matches the given initial distribution or boundary function. This turns the physical data into mathematical coefficients.
“Well-posedness” includes stability, which means
A Multiple solutions always
B Small data change small effect
C No solution ever
D Large data change small effect
Stability means small errors in initial or boundary data cause only small changes in the solution. This is essential for meaningful physical modeling and for reliable numerical computations.
In PDE language, “homogeneous equation” often means the equation has
A Zero source term
B Only mixed derivatives
C Only boundary values
D Only constant solutions
A homogeneous PDE has no forcing/source term, like u_t = k u_xx without added f(x,t). The solution behavior then depends purely on initial and boundary conditions.
A linear PDE allows superposition, meaning if u1 and u2 solve it then
A u1−u2 never solves
B u1·u2 always solves
C u1+u2 also solves
D only u1 solves
Linearity means the PDE operator is linear, so sums of solutions are also solutions. This property underlies Fourier series methods and building solutions by adding simpler modes.
A nonlinear PDE generally breaks superposition because it contains
A Only constant coefficients
B Only second derivatives
C Only boundary terms
D Products of u, derivatives
When terms like u·ux or (ux)² appear, adding two solutions usually does not give a new solution. This is why nonlinear PDEs need special methods and careful analysis.
The “total derivative” of F(x,y,z) along a curve means
A dF/dt with chain rule
B ∂F/∂x only
C ∂F/∂y only
D ∂F/∂z only
Along x(t), y(t), z(t), the total derivative uses the chain rule: dF/dt=Fx x′+Fy y′+Fz z′. This connects naturally with total differentials and characteristic ideas.
If Mdx+Ndy+Pdz is exact, then along any path between two points the integral depends on
A Path shape only
B Parameter choice only
C Endpoints only
D Time taken only
Exact differentials correspond to potential functions. The integral of dF from one point to another equals F(final)−F(initial), so it is path-independent, depending only on endpoints.
In a linear system, if eigenvalues are complex a±bi, solutions typically involve
A Only polynomial terms
B Exponential with sin/cos
C Only step functions
D Only constant solutions
Complex eigenvalues produce oscillatory behavior with exponential scaling. Using Euler’s formula, e^(at)(cos bt and sin bt) terms appear, giving rotating/oscillating solutions in phase space.
For dx/P=dy/Q=dz/R, two independent integrals are needed because
A PDE becomes second order
B Boundary is unnecessary
C Solution uses arbitrary F
D Degree becomes undefined
The general solution is F(u,v)=0, where u and v are independent first integrals. Without two independent integrals, you cannot build the full family of solution surfaces.
In classification, “characteristics” are curves where the PDE changes into
A ODE form
B Algebraic form
C Exact form only
D Fourier form
Characteristics convert PDE behavior into ordinary differential relations along special curves. This is especially central for first-order PDEs and also appears in hyperbolic second-order equations.
In a boundary value problem for Laplace’s equation, specifying u on boundary is called
A Neumann condition
B Dirichlet condition
C Initial condition
D Charpit condition
Dirichlet boundary conditions specify the value of the solution on the boundary. For Laplace’s equation, Dirichlet data often ensures a unique solution inside the region.
Specifying ∂u/∂n on the boundary is called
A Dirichlet condition
B Initial curve data
C Neumann condition
D Lagrange condition
Neumann conditions specify the normal derivative on the boundary, representing flux. For elliptic PDEs, pure Neumann problems may need an additional compatibility condition for uniqueness.
For Neumann boundary condition on Laplace’s equation, uniqueness is usually up to
A A sine function
B A multiplicative constant
C A quadratic polynomial
D An additive constant
If only normal derivatives are specified, adding a constant to a solution does not change its derivatives, so the boundary condition still holds. Hence uniqueness is typically up to a constant.
In a heat equation on a rod, boundary conditions usually represent
A End temperatures or flux
B Only eigenvalues
C Only discriminant
D Only exactness
Heat flow problems commonly specify temperature at the ends (Dirichlet) or heat flux at the ends (Neumann). These conditions, along with initial temperature, determine the temperature evolution.
In a wave equation on a string, fixed-end boundary conditions mean
A Velocity zero at ends
B Temperature zero at ends
C Displacement zero at ends
D Flux constant at ends
A fixed end cannot move, so the displacement u is zero at that boundary for all time. This boundary condition determines the allowed standing wave modes on the string.
A “particular solution” for a forced linear system is often found using
A Undetermined coefficients idea
B Discriminant test
C Exactness check
D Canonical transform
For simple forcing like exponentials or sines, one guesses a particular form with unknown constants and substitutes it into the system. Solving for constants yields a particular solution.
In total differential equations, the method of multipliers is different from integrating factor because it
A Always uses μ(x)
B Requires Fourier series
C Needs discriminant sign
D Builds integrals directly
Multipliers aim to create integrable combinations like ldx+mdy+ndz=0 that yield first integrals. Integrating factors instead multiply the entire form to make it exact.
For first-order PDE, “initial curve” data is important because it
A Changes PDE degree
B Selects unique solution
C Removes boundary needs
D Forces elliptic type
The general solution contains an arbitrary function. Initial curve data provides a specific relation that determines that function, resulting in a unique solution surface consistent with the given curve values.
In separation of variables, the spatial eigenfunctions for fixed ends are usually
A Sine functions
B Cosine only
C Exponential only
D Polynomial only
Fixed-end conditions require the function to be zero at endpoints. Sine functions naturally satisfy sin(0)=0 and sin(nπ)=0 at boundaries, making them the standard eigenfunctions.
For an insulated end in heat flow, the boundary condition is typically
A Zero temperature
B Zero time derivative
C Zero normal derivative
D Constant temperature
Insulation means no heat flux across the boundary. Heat flux is proportional to the spatial derivative, so the normal derivative is set to zero, which is a Neumann-type boundary condition.
In a PDE, “source term” typically represents
A External input effect
B Mixed derivative term
C Discriminant value
D Boundary geometry
A source term adds production or forcing inside the domain, such as heat generation. It changes the solution from the homogeneous case and must be included when matching physical behavior.
In a coupled ODE system, writing it as X′=AX is possible only if
A It is a PDE
B Coefficients are linear
C It is exact form
D It is separable always
Matrix form X′=AX is used for linear systems with variables appearing linearly and often with constant coefficients. Nonlinear terms prevent representing the system purely as a matrix multiplication.
In Lagrange PDE, if one chooses multipliers equal to (x,y,z), the goal is to get
A A discriminant value
B A boundary condition
C An integrable relation
D A Fourier coefficient
Choosing multipliers is a technique to form combinations of dx, dy, dz that integrate to a first integral. The choice depends on P, Q, R to simplify and integrate the resulting expression.
In PDE classification, converting to canonical form is most helpful because it
A Matches standard equations
B Removes all conditions
C Makes PDE nonlinear
D Guarantees exactness
Canonical forms resemble standard models like wave, heat, or Laplace equations. Once matched, you can apply known methods and understand which data (initial or boundary) is needed.
A typical reason Fourier series appear is because eigenfunctions form
A A random set
B A constant set
C A nonlinear set
D An orthogonal basis
Eigenfunctions from boundary value problems are orthogonal. This allows expanding initial/boundary data as a series in that basis, making coefficient calculation systematic and accurate.
In first-order PDE, “characteristic direction” at a point depends on
A Discriminant only
B Coefficients P and Q
C Boundary values only
D Eigenvalues of A
For Pp+Qq=R, the direction of characteristic projection in the x–y plane is governed by dx/P=dy/Q. Thus P and Q define the slope and direction field of characteristics.
In Lagrange PDE, if one integral is u(x,y,z)=c1, then along characteristics
A u remains constant
B u always increases
C u becomes zero
D u becomes linear
A first integral is constant along characteristic curves by definition. This means u does not change while moving along a characteristic, helping label solution surfaces and build general solutions.
In exact differential equations, the solution method mainly requires
A Finding discriminant
B Matrix eigenvalues only
C Building potential function
D Fourier expansion only
Exactness means Mdx+Ndy(+Pdz) equals dF. The main task is finding F by integration and consistency checks, then writing the implicit solution F = constant.
For hyperbolic PDE, characteristics matter most because they describe
A Only boundary geometry
B Only eigenfunction basis
C Only exactness condition
D Paths of information travel
Hyperbolic PDEs carry signals along characteristic curves, similar to wave motion. These curves show where data influences the solution, guiding how initial conditions determine future behavior.