Chapter 22: Differential Equations (PDE) and Total Differential Equations (Set-2)
In a coupled system where dx/dt depends on both x and y, the system is called
A Single separable ODE
B Pure algebraic set
C Coupled ODE system
D Second-order PDE
When derivatives of multiple dependent variables are linked together, the equations must be solved simultaneously. Such systems commonly model interacting quantities, like position–velocity pairs or two connected physical variables.
If two first-order ODEs are given, a standard reduction approach is to
A Reduce to one ODE
B Raise to third order
C Convert into Laplace form
D Expand as Taylor series
Many easy systems are solved by eliminating one variable to get one equation in a single unknown. After solving it, substitute back to obtain the other variable’s solution.
A “homogeneous linear” simultaneous system typically has
A Only quadratic terms
B Zero forcing terms
C Only constants present
D No derivatives involved
Homogeneous linear systems have no external input term on the right side. Solutions then depend on initial conditions and the system’s internal structure, often producing exponential-type behaviors.
Initial conditions in simultaneous differential equations are mainly used to
A Change system type
B Remove derivatives fully
C Fix constants uniquely
D Add extra variables
Solving a system usually gives a general solution containing constants. Initial conditions provide specific values of variables at a given point, allowing those constants to be determined uniquely.
When a system is written in symmetric form, it often helps because it
A Removes all unknowns
B Guarantees exactness
C Forces parabolic type
D Simplifies substitutions
Symmetry can suggest useful combinations like x+y or x−y that reduce the system. This often makes elimination and integration easier without changing the meaning of the original model.
In some simple linear systems, the eigenvalue idea is used to find
A Exponential solution forms
B Only polynomial roots
C Only constant solutions
D Only trigonometric series
For constant-coefficient linear systems, solutions are often combinations of exponentials. Eigenvalues guide which exponent rates appear, and eigenvectors describe how variables combine in each mode.
A total differential equation is commonly written using the form
A dy/dx = y/x
B uxx + uyy = 0
C Mdx+Ndy+Pdz=0
D p + q = 0
Total differential equations often appear as a sum of terms like Mdx, Ndy, and Pdz. The key task is checking integrability and then integrating to obtain an implicit relation.
If Mdx + Ndy + Pdz equals dF, then the equation becomes
A z = constant
B F = constant
C x = 0 only
D y = 0 only
If the expression is an exact differential dF, then dF=0 implies F does not change along solutions. Therefore the general solution is F(x,y,z)=C as an implicit form.
The “potential function” in a total differential setting refers to
A A single scalar F
B A boundary curve only
C A mixed derivative term
D A time-dependent forcing
A potential function F(x,y,z) is one whose total differential matches the given form. Existence of F means the equation is integrable, and solutions are obtained as F = constant.
An integrating factor is introduced mainly to
A Increase equation degree
B Remove independent variables
C Make form integrable
D Create a new PDE
When a differential form is not exact, multiplying by a suitable integrating factor can make it exact. Then it becomes a total differential of some function, enabling direct integration.
A basic “integrability condition” is used to decide whether
A Order is defined
B Degree is always one
C Boundary is required
D A potential exists
Integrability conditions check whether the differential form can be expressed as dF. If satisfied, integration gives F(x,y,z)=C. If not, one may try an integrating factor.
In thermodynamics language, exact differentials usually represent
A Path-dependent work
B Random heat transfer
C State functions change
D External forcing only
State functions depend only on the state, not the path, so their differentials are exact. This connects naturally with exact differential equations and potential functions in mathematics.
A first-order PDE involves derivatives like
A ∂z/∂x, ∂z/∂y
B ∂²z/∂x², ∂²z/∂y²
C Only ordinary derivative
D No derivative terms
First-order PDEs contain only first partial derivatives. They often describe how a surface changes with two independent variables and can be solved using characteristic curves for many simple forms.
In Lagrange’s linear PDE, the symbols p and q usually mean
A Second partial derivatives
B First partial derivatives
C Independent variables
D Boundary constants
Standard notation sets p=∂z/∂x and q=∂z/∂y. This simplifies writing the PDE and helps connect it to the auxiliary system used in the method of characteristics.
A Lagrange linear PDE is often expressed as
A uxx + uyy = 0
B y′ + Py = Q
C Pp + Qq = R
D dy/dx = f(x)
In this form, P, Q, and R are functions of x, y, z. The method uses auxiliary equations to build two independent integrals, then combines them into a general solution.
The auxiliary system for Lagrange’s PDE is written as
A dx/dy = P/Q only
B dy/dx = Q/P only
C dz/dx = R/P only
D dx/P = dy/Q = dz/R
These equal ratios define characteristic directions where the PDE becomes simpler. Solving them produces two independent integrals, which are then combined to form the general integral of the PDE.
In the method of multipliers, we choose multipliers mainly to
A Create an exact differential
B Increase mixed derivatives
C Force elliptic behavior
D Remove all constants
Multipliers are selected so that a linear combination of dx, dy, dz becomes integrable, giving a first integral. Repeating gives two integrals needed to construct the general solution.
A “general integral” of Lagrange PDE is commonly written as
A u + v = 0 only
B u = 0 always
C F(u, v) = 0
D v = 1 always
After finding two independent first integrals u=c1 and v=c2 from the auxiliary system, the general solution is expressed as an arbitrary relation F(u,v)=0.
A “complete integral” of a first-order PDE generally contains
A One arbitrary constant
B Two arbitrary constants
C No arbitrary constant
D Only fixed numbers
For typical first-order PDEs in two independent variables, a complete integral includes two parameters. These parameters can later be related using given conditions to obtain a particular solution.
A “particular integral” is usually obtained by
A Applying given conditions
B Raising PDE order
C Removing variables entirely
D Adding mixed derivatives
Conditions such as initial curves or specific constraints select special values or relations among parameters in the complete integral, producing a particular solution that fits the problem data.
Charpit’s method is mainly associated with
A Linear second-order PDE
B Exact total differentials
C Nonlinear first-order PDE
D Coupled linear ODE
Charpit’s method is designed for nonlinear first-order PDEs. It forms characteristic-like equations to obtain a complete integral, from which particular integrals may be derived using conditions.
A characteristic curve is useful because along it, the PDE becomes
A An ordinary differential equation
B A second-order algebraic
C A pure boundary condition
D A Fourier expansion
The method of characteristics converts a PDE into ODEs along special curves in the plane. Solving these ODEs helps build the surface that satisfies the original PDE.
A second-order PDE in x and y usually includes derivatives like
A ux, uy only
B u, x, y only
C z′ and z″ only
D uxx, uxy, uyy
Second-order PDEs contain second partial derivatives, typically uxx, uxy, and uyy with coefficients. These terms determine classification and strongly influence solution behavior and required conditions.
The discriminant used for classification is based on coefficients of
A ux and uy only
B u and x only
C uxx, uxy, uyy
D constants only
Classification uses the second-order part coefficients A, B, C. The discriminant (commonly B²−4AC in a standard form) decides whether the PDE is elliptic, parabolic, or hyperbolic.
A hyperbolic second-order PDE typically has
A No real characteristic
B Two real characteristics
C One repeated characteristic
D Only constant solutions
Hyperbolic PDEs have two distinct characteristic directions, matching wave-like propagation. This leads to solutions influenced strongly by initial data along curves and transport of information.
A parabolic second-order PDE typically has
A One repeated characteristic
B Two real characteristics
C No real characteristic
D Only algebraic curves
Parabolic PDEs have a repeated characteristic direction, typical of diffusion problems. This structure matches the heat equation type, where initial data spreads and smooths over time.
An elliptic second-order PDE typically has
A No real characteristics
B Two real characteristics
C One repeated characteristic
D Only time derivatives
Elliptic PDEs do not have real characteristic curves in the usual sense. They often model steady-state fields, and solutions are typically determined by boundary values on a region.
A standard example of an elliptic PDE is
A Heat equation type
B Laplace equation type
C Wave equation type
D Logistic growth ODE
The Laplace equation represents steady potential fields and is elliptic. Its solutions are strongly controlled by boundary conditions, and it commonly appears in electrostatics and fluid flow.
A standard example of a parabolic PDE is
A Wave equation type
B Laplace equation type
C Simple harmonic ODE
D Heat equation type
The heat equation models diffusion of temperature. It is parabolic, so initial data evolves smoothly with time, and boundary conditions on space are often required for uniqueness.
A standard example of a hyperbolic PDE is
A Heat equation type
B Laplace equation type
C Wave equation type
D Bernoulli ODE
The wave equation describes propagation of disturbances at finite speed. It is hyperbolic, with two characteristic families, and typically uses initial displacement and initial velocity conditions.
Boundary value problems are most commonly linked with
A Elliptic PDE models
B Only coupled ODEs
C Only exact ODEs
D Only nonlinear PDEs
Elliptic problems often require boundary conditions on a closed region to determine a solution. This matches steady-state physical situations where values are prescribed along boundaries.
Initial value problems are especially common for
A Only Laplace PDE
B Only total differentials
C Heat and wave PDE
D Only exact systems
Heat and wave equations describe time evolution, so initial conditions at a starting time are essential. Boundary conditions may also be present, but initial data drives the future behavior.
A linear PDE means the dependent variable appears
A Always squared terms
B Linearly with derivatives
C Only as a constant
D Without any derivatives
Linearity means the unknown function and its derivatives occur only to the first power and are not multiplied together. This allows superposition and makes many solution methods simpler.
A nonlinear PDE may contain terms like
A uxx + uyy only
B ux + uy only
C u + x + y only
D (ux)(uy) term
Products or powers of the unknown or its derivatives create nonlinearity. Such equations usually cannot use superposition, and solution methods become more specialized.
The order of a PDE is determined by
A Number of variables
B Size of coefficients
C Highest derivative present
D Number of constants
Order equals the highest-order partial derivative in the equation. For example, any appearance of second partial derivatives makes the PDE second order, regardless of additional lower-order terms.
The degree of a PDE is defined only when derivatives appear
A Polynomially in equation
B Inside sine functions
C Inside exponentials
D Under square roots
Degree is meaningful only if the PDE is a polynomial in derivatives. If derivatives occur in non-polynomial ways, like sin(ux) or e^(ux), the degree is not defined.
In second-order PDEs, the mixed derivative term is
A uxx term present
B uxy term present
C uyy term present
D ux term present
The mixed derivative uxy couples x and y variation directly. Its coefficient affects classification through the discriminant and can sometimes be removed or simplified using a change of variables.
Canonical transformation is mainly used to
A Increase equation degree
B Add extra variables
C Simplify second-order part
D Make solution numerical
Canonical transformations change variables so the second-order portion becomes simpler, revealing the PDE type clearly. This helps reduce complicated forms to standard elliptic/parabolic/hyperbolic models.
Separation of variables is most suitable when the PDE is
A Fully nonlinear always
B Random coefficient only
C Exact total differential
D Linear with simple BC
Separation assumes a product form like X(x)T(t). It works best for linear PDEs with boundary conditions that fit eigenfunction expansions, such as heat and wave problems on intervals.
Fourier series is commonly used to satisfy
A Boundary conditions neatly
B Exactness conditions only
C Elimination in systems
D Charpit equations
Fourier series represent functions as sums of sines and cosines that naturally match boundary constraints. In PDE solutions, they often appear after separation of variables to fit given boundaries.
A “well-posed” PDE problem generally requires
A Only uniqueness condition
B Only existence condition
C Existence, uniqueness, stability
D Only stability condition
A well-posed problem has a solution that exists, is unique, and changes continuously with small changes in data. This ensures physical meaning and reliable numerical approximation.
Homogeneous boundary condition usually means the boundary value is
A Constant nonzero value
B Zero on boundary
C Unknown by choice
D Infinite everywhere
Homogeneous boundary conditions set the function (or specified derivative) to zero along the boundary. This simplifies solutions, especially in separation of variables, by removing extra constant terms.
In a total differential equation, a common quick check is to
A Test for exactness
B Solve by separation
C Apply Fourier series
D Classify by discriminant
Before integrating, you check whether the differential form behaves like dF. If exact, integrate directly. If not, search for an integrating factor that makes integration possible.
If a form is exact, the solution is found mainly by
A Finding eigenvalues first
B Using Charpit method
C Direct integration to F
D Computing discriminant
Exactness means the expression equals dF. Integrating gives a potential function F, and the solution becomes F(x,y,z)=C. This is one of the simplest cases in differential forms.
In Lagrange PDE, “two integrals” are needed because the solution uses
A A fixed constant only
B Only one parameter
C No free choice
D An arbitrary function
Two independent integrals u and v allow the general solution F(u,v)=0, where F is arbitrary. This arbitrary function represents a family of solution surfaces fitting the PDE.
A physical interpretation often linked with hyperbolic PDE is
A Steady potential behavior
B Signal propagation behavior
C Pure diffusion smoothing
D Random noise filtering
Hyperbolic equations allow disturbances to travel along characteristics, like waves. This matches real signal propagation where effects move with finite speed rather than spreading instantly everywhere.
A physical interpretation often linked with parabolic PDE is
A Undamped wave motion
B Diffusion and smoothing
C Steady equilibrium field
D Exactness of forms
Parabolic PDEs model diffusion processes where peaks flatten with time. Heat flow is the classic example, showing gradual spreading and smoothing under initial and boundary constraints.
A physical interpretation often linked with elliptic PDE is
A Steady-state equilibrium
B Time-evolving diffusion
C Traveling wave signals
D Coupled motion system
Elliptic PDEs commonly describe equilibrium states with no time evolution, like steady temperature distribution or electric potential. Boundary values control the entire interior solution strongly.
For dx/P = dy/Q = dz/R, obtaining one first integral usually means finding
A A unique numeric answer
B A boundary-only formula
C One relation among x,y,z
D A second-order derivative
A first integral is an expression u(x,y,z)=C that stays constant along characteristics. Finding two independent such relations lets you build the general solution of the PDE.
When solving a simple coupled linear system, after reduction you finally
A Change PDE classification
B Substitute back to get other
C Replace with boundary values
D Remove initial conditions
After forming and solving one ODE for a variable, you use the original system to compute the second variable. This back-substitution ensures both equations are satisfied together.