Chapter 22: Differential Equations (PDE) and Total Differential Equations (Set-1)
A “simultaneous differential equation” usually means
A One separable ODE
B One exact ODE
C Two coupled ODEs
D One linear PDE
Simultaneous differential equations are systems where two or more differential equations involve the same dependent variables, so they must be solved together, often by elimination or reduction to a single equation.
A common first step to solve two coupled ODEs is
A Eliminate a variable
B Differentiate twice only
C Replace with a series
D Apply Laplace always
Many basic systems are solved by eliminating one variable to form a single differential equation in the other variable, solve it, then substitute back to obtain the remaining solution.
The notation dx/P = dy/Q = dz/R represents
A Bernoulli equation form
B Lagrange auxiliary system
C Exact ODE in x
D Euler–Cauchy equation
In first-order linear PDE (Lagrange’s form), characteristic/auxiliary equations are written as dx/P = dy/Q = dz/R, helping to find two independent integrals to build the solution.
A “total differential equation” commonly appears as
A dy/dx + y=0
B ∂u/∂x = 0 only
C y″ + y=0
D Mdx+Ndy+Pdz=0
Total differential equations often use the Pfaffian form Mdx+Ndy+Pdz=0. The goal is to check integrability and find a potential function giving an implicit solution.
An equation is “exact” when it equals
A dF = 0 form
B dy/dx = 0
C ∂²u/∂x² only
D y = constant only
Exactness means the differential expression is the total differential of some function F(x,y,…) so the equation becomes dF=0 and integrates directly to F = constant.
In many problems, an integrating factor is used to
A Increase equation order
B Remove all constants
C Make equation exact
D Change variables only
If a differential form is not exact, multiplying by a suitable integrating factor can convert it into an exact differential, allowing integration to a potential function and an implicit solution.
A “potential function” in total differential form means
A F exists with dF
B Only x is present
C Only y is present
D No derivatives exist
If Mdx+Ndy+Pdz equals dF for some F(x,y,z), then F is a potential function. The solution is found as F(x,y,z)=C after integration.
A first-order PDE involves
A Second partial derivatives
B Only ordinary derivatives
C Only algebraic terms
D First partial derivatives
A first-order PDE contains only first partial derivatives like ∂z/∂x and ∂z/∂y (or similar). Methods like characteristics are commonly used to solve such PDEs.
Lagrange’s linear PDE is usually written as
A y′+Py=Q
B y″+Py′=0
C Pp+Qq=R
D ∂²u/∂x²=0
In Lagrange’s linear PDE, p=∂z/∂x and q=∂z/∂y, giving the standard form Pp+Qq=R. It is solved using auxiliary (characteristic) equations.
In Lagrange’s method, we find solutions by
A Two independent integrals
B One constant only
C Fourier series first
D Laplace transform only
The auxiliary system yields two independent integrals u(x,y,z)=c1 and v(x,y,z)=c2. The general solution is then F(u,v)=0, where F is arbitrary.
A “complete integral” in first-order PDE usually contains
A No parameters
B Two parameters
C One parameter
D Infinite parameters
For first-order PDE in two variables, a complete integral generally contains two arbitrary constants (parameters). From it, one can derive particular solutions by imposing conditions.
A “particular integral” is obtained by
A Raising PDE order
B Removing all constants
C Fixing parameters
D Making PDE nonlinear
A particular integral is obtained by selecting specific values or relations among the parameters in the complete integral, often using initial/boundary conditions to fit a given curve or data.
“Characteristic curves” are used mainly to
A Reduce PDE to ODE
B Increase variables count
C Create new PDE always
D Remove derivatives fully
Along characteristic curves, a first-order PDE becomes an ordinary differential equation. Solving along these curves provides integral surfaces that represent the PDE solution.
The degree of a PDE is defined only when
A Linear in variables
B Always second order
C Always homogeneous
D Polynomial in derivatives
Degree is defined when the PDE can be expressed as a polynomial in its partial derivatives. If derivatives appear inside functions like sin or exp, degree is not defined.
The order of a PDE is
A Number of variables
B Highest derivative order
C Number of constants
D Coefficient magnitude
Order is determined by the highest-order partial derivative present. For example, if ∂²u/∂x² appears, the PDE is second order regardless of other lower derivatives.
A general second-order PDE in x,y has form
A p+q=0 only
B y′+y=0
C Auxx+2Buxy+Cuyy+…
D dy/dx = f(x)
The standard second-order PDE involves second derivatives uxx, uxy, uyy with coefficients A, B, C plus lower-order terms. Classification depends on the discriminant B²−AC (or B²−4AC by convention).
PDE classification uses the sign of
A Discriminant term
B Constant term only
C First derivative only
D Initial condition only
For a second-order PDE, classification is based on the discriminant (commonly B²−4AC or B²−AC depending on notation). Its sign decides hyperbolic, parabolic, or elliptic type.
If the discriminant is positive, the PDE is
A Elliptic
B Parabolic
C Hyperbolic
D Algebraic
A positive discriminant indicates two distinct real characteristic directions, typical of wave-like behavior. Hence the PDE is classified as hyperbolic, like the standard wave equation.
If the discriminant is zero, the PDE is
A Hyperbolic
B Elliptic
C Nonlinear
D Parabolic
Zero discriminant corresponds to repeated characteristic direction, typical of diffusion/heat flow behavior. Therefore the PDE is parabolic, like the heat equation.
If the discriminant is negative, the PDE is
A Elliptic
B Hyperbolic
C Parabolic
D Separable ODE
Negative discriminant implies no real characteristic curves, typical of steady-state potential problems. The PDE is elliptic, like Laplace’s equation, often linked to boundary value problems.
The classic example of a hyperbolic PDE is
A Heat equation
B Wave equation
C Laplace equation
D Logistic equation
The wave equation models propagation of waves and has two real characteristic families. This behavior matches hyperbolic classification, where signals travel along characteristic curves.
The classic example of a parabolic PDE is
A Wave equation
B Laplace equation
C Heat equation
D Bernoulli equation
The heat equation describes diffusion-like processes and has a repeated characteristic direction, which matches parabolic classification. Solutions are typically governed by initial conditions evolving over time.
The classic example of an elliptic PDE is
A Laplace equation
B Heat equation
C Wave equation
D Exact ODE
Laplace’s equation models steady-state fields like electrostatic potential. It has no real characteristics, fitting elliptic type, and solutions are usually determined by boundary conditions.
Boundary conditions are most naturally tied to
A Only simultaneous ODEs
B Elliptic problems
C Only exact ODEs
D Only separable ODEs
Elliptic PDEs like Laplace’s equation typically require boundary values on a closed region to determine a unique solution, making boundary value problems central for elliptic equations.
Initial conditions are most common with
A Elliptic only
B Algebraic only
C Exact only
D Hyperbolic/parabolic
Hyperbolic (wave) and parabolic (heat) PDEs often describe time evolution, so initial conditions at time t=0 are essential. Boundary conditions may also appear, but initial data is key.
In dx/P = dy/Q = dz/R, P,Q,R are
A Constant numbers only
B Always zero terms
C Coefficients functions
D Boundary values
P, Q, R are functions of x, y, z that come from the PDE Pp+Qq=R. They define the characteristic system used to compute integrals and construct the general solution.
A “linear PDE” means it is linear in
A Unknown and derivatives
B Only independent variables
C Only constants present
D Only boundary values
A PDE is linear if the dependent variable and its derivatives appear only to the first power and are not multiplied together. Coefficients may depend on independent variables.
A “nonlinear PDE” may include
A Only uxx term
B Product of derivatives
C Only constant coefficients
D No dependent variable
Nonlinearity occurs when the unknown or derivatives appear multiplied (like (ux)(uy)) or raised to powers. Such PDEs are generally harder and may not follow superposition.
Separation of variables is mainly used for
A Only coupled ODEs
B Only exact equations
C Linear PDE with BC
D Only algebraic PDE
Separation of variables assumes a product form like u=X(x)T(t) and is widely used for linear PDEs such as heat/wave/Laplace equations, especially with boundary conditions on intervals.
Fourier series is most often linked with
A PDE boundary solutions
B Exactness testing
C Eliminating variables
D Charpit method
Fourier series expansions help represent solutions that satisfy boundary conditions in separation-of-variables methods, especially for heat and wave equations on finite domains.
A system called “homogeneous” often means
A No derivatives present
B No variables present
C No constants allowed
D No forcing terms
In many linear systems, “homogeneous” means the right-hand side is zero (no external input). This typically leads to solutions determined by initial conditions and system structure.
A simple way to reduce two ODEs is
A Differentiate one equation
B Ignore one variable
C Replace by PDE
D Add random constants
If one equation relates derivatives, differentiating and substituting from the other can eliminate a variable. This converts the system into a single higher-order ODE, then back-substitute.
“Initial conditions” help to
A Change PDE type
B Define degree
C Fix constants
D Remove variables
General solutions contain constants or arbitrary functions. Initial conditions provide specific values at a starting point or curve, which determines those constants and gives a unique solution.
In total differential equations, “verification” often means
A Compute Laplace only
B Check exactness condition
C Find series solution
D Assume linear always
You typically verify whether the given differential form is integrable/exact by checking appropriate conditions. If not exact, you may search for an integrating factor to make it exact.
The “method of multipliers” is used to
A Raise PDE order
B Reduce variables count
C Delete mixed term
D Create exact combination
In Lagrange’s PDE, choosing multipliers l, m, n so that ldx+mdy+ndz becomes integrable helps find first integrals. This is called using multipliers to obtain solutions.
A “Pfaffian form” commonly refers to
A Mdx+Ndy+Pdz
B y′ = f(x)
C uxx + uyy
D p + q = 0
Pfaffian form is a general differential expression involving multiple differentials. Solving it often involves testing integrability and finding a potential function or an integrating factor.
In thermodynamics, exact differentials are linked to
A Path functions only
B Random variables
C State functions
D Boundary curves
State functions like internal energy depend only on the state, so their differentials are exact. Path-dependent quantities like heat and work are not exact differentials in general.
“Characteristic equations” for Lagrange PDE are
A dy/dx = y/x
B dx/P=dy/Q=dz/R
C y″ + y = 0
D uxx + uyy = 0
Lagrange’s PDE uses auxiliary equations dx/P = dy/Q = dz/R. Solving these yields integrals that define the characteristic curves/surfaces, forming the PDE’s general solution.
A “general integral” of first-order PDE is often
A F(u,v)=0
B u+v=1 only
C u=v always
D u=0 only
After finding two independent integrals u=c1 and v=c2 of the characteristic system, the general integral is expressed as F(u,v)=0 where F is an arbitrary function.
“Canonical form” aims to
A Increase PDE degree
B Remove all constants
C Simplify PDE structure
D Make it nonlinear
Transforming a PDE to canonical form via change of variables simplifies the second-order part and reveals the PDE type clearly. This helps in analysis and in selecting suitable solution methods.
A “well-posed” PDE problem generally needs
A Only uniqueness
B Only existence
C Only stability
D Existence, uniqueness, stability
A well-posed problem has a solution that exists, is unique, and depends continuously on data. These ideas are crucial in PDEs to ensure meaningful physical and numerical solutions.
In PDEs, “homogeneous” boundary conditions mean
A Boundary value unknown
B Boundary value is zero
C Boundary is curved
D Boundary is infinite
Homogeneous boundary conditions mean the dependent variable (or its derivative, depending on type) is zero on the boundary. This often simplifies separation-of-variables solutions.
“Linear simultaneous equations” in systems means
A Linear in unknowns
B Linear in time only
C Quadratic in unknowns
D No unknowns exist
A linear system has the dependent variables and their derivatives appearing linearly (no products or powers). Such systems are usually handled by elimination, substitution, or matrix methods.
A “symmetric form system” often shows
A Different variables only
B No derivatives
C Similar structure in equations
D Only constants
Symmetry in a system means equations look similar under variable exchange, which sometimes allows adding/subtracting equations or using substitutions to reduce the system neatly.
A typical goal in PDE classification is to identify
A Elliptic/parabolic/hyperbolic
B Only exactness
C Only integrating factor
D Only eigenvalues
Classification predicts behavior and the right type of conditions needed. Hyperbolic relates to waves, parabolic to diffusion, elliptic to steady-state fields, guiding methods and interpretations.
Mixed derivative term in second-order PDE is
A uxx term
B uyy term
C ux term
D uxy term
The mixed derivative is the cross partial derivative uxy (or ∂²u/∂x∂y). Its coefficient affects the discriminant and can be reduced by a suitable change of variables.
“Auxiliary equations” are mainly used in
A Second-order elliptic only
B First-order linear PDE
C Only exact ODE
D Only Fourier series
Auxiliary equations are a key part of Lagrange’s first-order linear PDE method. They define characteristic curves along which the PDE becomes simpler to integrate.
In a total differential form, an implicit solution looks like
A y = mx + c
B u = X+T only
C F(x,y,z)=C
D p = q only
When a differential form is exact, integrating produces a potential function F. Setting dF=0 gives F(x,y,z)=constant, which is the implicit general solution.
A basic meaning of “integrability condition” is
A Condition for exactness
B Condition for degree
C Condition for order
D Condition for boundary
Integrability conditions ensure a differential form can be integrated to a single-valued function. In practice, they test whether a potential function exists, possibly after applying an integrating factor.
A PDE with time evolution and diffusion behavior is likely
A Elliptic type
B Hyperbolic type
C Parabolic type
D Algebraic type
Diffusion-like processes smooth out variations over time, typical of parabolic PDEs such as the heat equation. The discriminant condition indicates parabolic behavior and guides initial-boundary setup.