Chapter 22: Differential Equations (PDE) and Total Differential Equations (Set-3)
If a coupled linear system is written as dX/dt = AX, the usual solution idea uses
A Only separation
B Trial polynomial only
C Eigenvalues of A
D Fourier series only
Writing a linear system in matrix form dX/dt=AX allows solutions built from eigenvalues and eigenvectors. Each eigenvalue gives an exponential mode, and constants are fixed using initial conditions.
For two coupled first-order ODEs, elimination often produces
A A second-order PDE
B A pure algebraic set
C A Laplace equation
D One higher-order ODE
Eliminating one variable from two first-order equations commonly leads to a single second-order ODE in the remaining variable. After solving it, substitute back to recover the eliminated variable.
A “nonhomogeneous” linear simultaneous system usually means
A No derivatives present
B Forcing term present
C Only constant solutions
D Exact system guaranteed
Nonhomogeneous systems include an external input term like dX/dt=AX+F(t). The total solution is the sum of the homogeneous solution and a particular solution caused by the forcing.
If a system has a particular solution plus homogeneous solution, this idea relies on
A Degree definition
B Parabolic condition
C Linearity property
D Exactness condition
For linear systems, the superposition principle holds: the general solution equals homogeneous solution plus any one particular solution. This breaks the problem into simpler parts and is widely used.
In a total differential equation Mdx+Ndy=0, exactness requires
A ∂M/∂y = ∂N/∂x
B ∂M/∂x = ∂N/∂y
C M = N always
D M and N constants
For a two-variable differential form Mdx+Ndy to be exact, mixed partial derivatives of the potential must match: ∂M/∂y equals ∂N/∂x. Then Mdx+Ndy=dF and integrates to F=C.
For Mdx+Ndy+Pdz=0, “integrable” mainly means
A It becomes uxx
B It is second order
C It is homogeneous
D It equals dF
Integrability here means there exists a function F(x,y,z) such that dF=Mdx+Ndy+Pdz. Then the equation dF=0 gives the implicit solution F(x,y,z)=constant.
A common method to force exactness is to multiply by
A Integrating factor
B Discriminant term
C Boundary value
D Eigenvector only
If a differential form is not exact, multiplying by a carefully chosen integrating factor μ can make it exact. After that, integration gives a potential function and a clear implicit solution.
In Lagrange PDE Pp+Qq=R, the characteristic curves are found from
A uxx+uyy=0
B dy/dx = y
C dx/P=dy/Q
D d²y/dx² only
The auxiliary system dx/P=dy/Q=dz/R defines characteristic directions. Using dx/P=dy/Q first gives characteristic curves in the x–y plane, which help build solution surfaces for the PDE.
If p=∂z/∂x and q=∂z/∂y, then Pp+Qq=R is
A Second-order PDE
B First-order linear PDE
C Exact ODE
D Separable ODE
The equation involves only first partial derivatives p and q and is linear in them. Such PDEs are solved using Lagrange’s method and characteristic (auxiliary) equations.
The general solution of Lagrange PDE is often written as
A F(u,v)=0
B u=v only
C z=x+y always
D z=constant only
Two independent first integrals u(x,y,z)=c1 and v(x,y,z)=c2 are found from the auxiliary system. The general integral is then any relation F(u,v)=0.
A “complete integral” in first-order PDE normally depends on
A One parameter
B No parameter
C Two parameters
D Only boundary values
For first-order PDEs in two independent variables, a complete integral typically contains two arbitrary constants. These constants allow generating a family of solutions and fitting conditions later.
Charpit’s method is mainly applied when the first-order PDE is
A Linear type
B Second order
C Exact form
D Nonlinear type
Charpit’s method handles nonlinear first-order PDEs by forming a system of characteristic relations. From these, a complete integral can be obtained and then adjusted to meet given conditions.
A “particular integral” is selected from the complete integral using
A Discriminant sign
B Given condition
C Exactness test
D Fourier expansion
A particular solution is obtained by applying an initial curve or constraint to relate the parameters in the complete integral. This reduces the family to the one solution that matches the data.
The order of a PDE is based on the
A Number of constants
B Number of variables
C Highest derivative order
D Magnitude of coefficients
Order is the highest order of partial derivative present. If second derivatives like uxx occur, the PDE is second order even if first derivatives and function terms also appear.
The degree of a PDE is meaningful only if the PDE is
A Polynomial in derivatives
B Linear in variables
C Always homogeneous
D Always separable
Degree is defined when derivatives appear algebraically as a polynomial. If derivatives appear inside functions like sin(ux) or e^(ux), then degree cannot be defined in the usual sense.
A standard second-order PDE written as A uxx + 2B uxy + C uyy + … is classified using
A A+B+C sum
B Only C value
C B²−AC sign
D Only A value
For the form A uxx + 2B uxy + C uyy, classification depends on the discriminant B²−AC. Positive gives hyperbolic, zero gives parabolic, and negative gives elliptic type.
If B²−AC > 0, the PDE type is
A Hyperbolic
B Elliptic
C Parabolic
D Exact
A positive discriminant indicates two distinct real characteristic directions. This is typical of wave propagation problems, so the PDE is hyperbolic, like the standard wave equation.
If B²−AC = 0, the PDE type is
A Hyperbolic
B Parabolic
C Elliptic
D Linear ODE
Zero discriminant means repeated characteristics. This is typical of diffusion behavior, so the PDE is parabolic, like the heat equation, where initial data smooths out over time.
If B²−AC < 0, the PDE type is
A Hyperbolic
B Parabolic
C Elliptic
D Separable ODE
A negative discriminant implies no real characteristic directions. This matches steady-state field problems like Laplace’s equation, where boundary values control the solution in the region.
A canonical transformation is used mainly to
A Remove mixed term
B Add more variables
C Increase PDE order
D Force exactness
By changing variables, many second-order PDEs can be transformed to a canonical form that eliminates or simplifies the mixed derivative term, making the equation type and solution approach clearer.
A key difference: elliptic PDE problems usually emphasize
A Initial velocity only
B Time evolution only
C Integrating factor only
D Boundary conditions
Elliptic equations describe equilibrium states, so solutions are typically determined by boundary values on the region. Unlike time-evolution problems, initial data is not the main requirement.
A wave-type PDE is commonly associated with
A Parabolic class
B Elliptic class
C Hyperbolic class
D Exact class
Wave equations have two families of real characteristics, meaning signals travel along characteristic lines. This is the typical behavior of hyperbolic PDEs and guides the need for initial data.
A heat-type PDE is commonly associated with
A Parabolic class
B Hyperbolic class
C Elliptic class
D Bernoulli class
Heat equations model diffusion. Their repeated characteristic structure matches parabolic classification, and solutions depend strongly on initial conditions plus spatial boundary conditions for uniqueness.
Laplace equation corresponds to which class
A Parabolic class
B Elliptic class
C Hyperbolic class
D Exact class
Laplace’s equation describes steady potentials with no time dependence. It is elliptic and is typically solved as a boundary value problem where boundary values determine the interior solution.
In separation of variables for PDE, one assumes solution like
A Sum of constants
B Only linear z
C Product of functions
D Only polynomial x
Separation of variables assumes a form like u(x,t)=X(x)T(t). Substitution splits the PDE into ODEs with a separation constant, which are solved and then matched to boundary/initial data.
Fourier series in PDEs is often used to
A Fit boundary shapes
B Check exactness only
C Remove mixed derivatives
D Find eigenvalues only
Fourier series represent boundary or initial data as sums of sine/cosine modes. These modes naturally satisfy common boundary conditions and combine to form the full PDE solution.
A “well-posed” problem requires small data changes cause
A Large solution changes
B Small solution changes
C No solution exists
D Multiple solutions always
Stability means small changes in initial or boundary data lead to small changes in the solution. Along with existence and uniqueness, this forms the standard requirement for well-posed PDE problems.
A boundary condition is called homogeneous when it sets
A Value equal one
B Value unknown
C Value equal zero
D Value infinite
Homogeneous boundary conditions set the function or specified derivative to zero on the boundary. This simplifies the separated solutions and often removes extra constant terms in series forms.
A first-order PDE becomes simpler along characteristics because
A Derivatives combine nicely
B Degree becomes zero
C Order becomes zero
D Boundary disappears
Along characteristic curves, the PDE reduces to ODE relations where changes in x, y, and z are linked. This converts a partial derivative problem into ordinary integration steps.
The geometric idea of a first-order PDE solution is
A A straight line only
B A single point only
C A constant table
D A surface in space
A solution z(x,y) defines a surface. Characteristics are curves on this surface along which the PDE becomes an ODE, helping build the surface from simpler curve relations.
In simultaneous linear systems, an integrating factor idea is closest to
A Classifying PDE type
B Finding discriminant sign
C Solving linear equation type
D Removing mixed term
Integrating factors are used to solve certain linear differential equations by making them directly integrable. In systems, similar linear-solving ideas appear when reducing to a solvable linear ODE.
A parametric solution in a simple system usually expresses variables in terms of
A Only x
B A parameter t
C Only y
D Only constants
Parametric solutions describe x and y as functions of a parameter (often time). This is common in motion applications where two linked quantities evolve together with respect to one parameter.
In motion models, simultaneous differential equations often relate
A Position and velocity
B Area and volume
C Only temperature values
D Only boundary curves
Many basic motion systems couple position and velocity through derivatives, for example dx/dt=v and dv/dt=a(x,t). Such coupling naturally forms a simultaneous system needing joint solution.
The “method of multipliers” in Lagrange PDE helps to
A Find discriminant sign
B Create Fourier terms
C Increase PDE order
D Find first integrals
Multipliers are chosen so that a combination ldx+mdy+ndz becomes integrable, giving one first integral. A second independent integral then leads to the general solution F(u,v)=0.
In total differential equations, “verification step” usually means
A Expand Fourier series
B Compute eigenvalues
C Differentiate and compare
D Set boundary to zero
After finding a potential function F, you verify by differentiating F to check whether its differential matches Mdx+Ndy(+Pdz). This confirms correctness and catches missed integrating factors.
A key idea in “reduction to exact equation” is to
A Multiply by suitable μ
B Replace x by 1/x
C Increase variables count
D Differentiate repeatedly
If a form is not exact, an integrating factor μ(x,y,…) may exist so that μMdx+μNdy+… becomes exact. Then it integrates to μ-adjusted potential function and implicit solution.
In classification, characteristic curves are most closely linked to
A Elliptic type only
B Hyperbolic/parabolic types
C Total differentials only
D Coupled ODE only
Hyperbolic and parabolic PDEs have real characteristic directions, which guide propagation or diffusion behavior. Elliptic PDEs typically do not have real characteristic curves in the same way.
A second-order PDE is linear if
A only A,B,C constant
B only uxx appears
C u and derivatives linear
D boundary values known
Linearity requires the dependent variable and all derivatives appear to first power and are not multiplied together. Coefficients may depend on x and y, yet the PDE can still be linear.
A typical “boundary value” statement specifies values on
A Region boundary curve
B Initial time only
C Whole plane always
D A single point only
Boundary conditions give values of the solution or its derivative along the boundary of a region. This is essential for elliptic PDEs and also used with heat/wave problems on finite domains.
A typical “initial curve condition” in first-order PDE means
A Values on a curve
B Values at infinity
C Values on boundary only
D Values nowhere given
For first-order PDEs, initial data may be given along a curve in the plane. The solution surface is then constructed by extending from that curve along characteristic curves.
The phrase “exact system idea” in coupled ODEs suggests
A System is second order
B Derivatives match potential
C System is always nonlinear
D System needs Fourier
In some systems, relations can behave like a total differential where a potential-type function exists. Recognizing such structure can simplify solving by integrating to an implicit constant relation.
In Lagrange PDE, if P and Q are both zero at a point, it may cause
A Hyperbolic conversion
B Exactness guarantee
C Characteristic degeneracy
D Fourier expansion
Characteristics rely on ratios dx/P and dy/Q. If P and Q vanish, the direction field becomes undefined, requiring special handling or alternate integrals, since standard characteristic construction breaks down.
For a PDE, “order and degree” are both defined only when
A boundary is homogeneous
B equation is hyperbolic
C Fourier series exists
D Derivatives are algebraic
Order always uses highest derivative order, but degree needs the PDE written as a polynomial in derivatives. If derivatives appear non-algebraically, degree is not defined in the usual way.
A “canonical form summary” is mainly used to
A Compare with standard models
B Fix initial constants only
C Remove boundary data
D Make PDE nonlinear
Canonical forms convert PDEs into standard elliptic/parabolic/hyperbolic models. This helps identify correct boundary/initial conditions and choose methods like separation of variables or characteristic analysis.
In separation of variables, the separation constant usually comes from
A forcing term removal
B exactness condition
C Setting ratio equal constant
D discriminant condition
After substituting u=X(x)T(t), the PDE often becomes a sum of a pure x-term and a pure t-term. Each side must equal a constant, called the separation constant, producing ODEs.
In wave equation solutions, boundary conditions often determine
A Exactness of PDE
B Allowed mode shapes
C Discriminant sign
D Order of equation
Boundary conditions restrict which eigenfunctions are permitted. These become mode shapes (sine/cosine patterns) in the separated solution, and the full solution is a sum of permitted modes.
In heat equation solutions, initial condition mainly determines
A Fourier coefficients
B PDE classification
C Mixed derivative term
D Degree of PDE
After separation, the solution becomes a series of eigenfunctions. The initial temperature distribution is expanded in that eigenfunction basis, which fixes the Fourier coefficients and fully determines the solution.
“Linear vs nonlinear PDE” difference is most clearly seen in
A Order becomes different
B Degree becomes negative
C Superposition works or not
D Boundary disappears
For linear PDEs, sums of solutions remain solutions, so superposition applies. Nonlinear PDEs generally do not allow this, which is a major reason they are harder to solve.
In solving coupled ODEs, “reduction to single ODE” is useful because
A Standard ODE tools apply
B PDE methods needed
C boundary conditions vanish
D degree becomes zero
Once the system is reduced to a single ODE in one variable, familiar methods like integrating factor, characteristic equation, or separation become usable. Then the second variable is recovered by substitution.
In PDE classification, identifying the type first helps mainly to choose
A Integrating factor always
B Elimination always
C Only parametric form
D Correct conditions and method
PDE type predicts behavior: elliptic needs boundary data, parabolic needs initial plus boundary, hyperbolic needs initial data along characteristics. Type also suggests methods like separation or characteristics.