In second-order linear ODE theory, what is the Wronskian of two functions?
A A determinant test
B A derivative rule
C An integral constant
D A slope formula
The Wronskian is a determinant built from two functions and their first derivatives. It is used to test linear independence of solutions in second-order linear differential equations.
If the Wronskian of two solutions is nonzero on an interval, the solutions are
A Linearly dependent
B Always periodic
C Linearly independent
D Always bounded
For solutions of a linear homogeneous ODE, a nonzero Wronskian on an interval indicates the solutions are linearly independent, forming a fundamental set for the general solution.
If two solutions of a linear homogeneous ODE are linearly dependent, their Wronskian is
A Always negative
B Always increasing
C Always zero
D Always one
Linear dependence means one solution is a constant multiple of the other. This makes the Wronskian determinant zero everywhere on the interval where both functions are defined.
For a second-order linear ODE, a “fundamental set” usually contains
A Two independent solutions
B One solution
C Three solutions
D Only constants
A second-order linear homogeneous ODE has a two-dimensional solution space. Two linearly independent solutions form a fundamental set and generate the complete general solution.
Abel’s formula mainly gives information about
A Exactness test
B Wronskian behavior
C Laplace inversion
D Series convergence
Abel’s formula relates the Wronskian of solutions of a second-order linear ODE to the coefficient of y′. It helps determine how the Wronskian changes across an interval.
Which statement best matches the superposition principle?
A Product of solutions works
B Inverse of solution works
C Sum of solutions works
D Only constant works
In linear homogeneous ODEs, any linear combination of solutions is also a solution. This superposition principle is why we build general solutions using independent solutions.
In p-notation, p commonly represents
A dy/dx
B y/x
C d²y/dx²
D ∫y dx
In first-order differential equations, p is often used as shorthand for dy/dx. This notation is common in equations solvable for x, y, or p.
An equation “solvable for p” means it can be written as
A y = f(p)
B x = f(y)
C p = constant
D p = f(x,y)
“Solvable for p” means dy/dx can be explicitly expressed in terms of x and y. Then standard techniques may be applied to find or analyze solutions.
In equations “solvable for x,” the form typically is
A y = f(x,p)
B p = f(x,y)
C x = f(y,p)
D y = constant
If a first-order equation is solvable for x, it can be rearranged to express x explicitly using y and p=dy/dx, often enabling parametric or elimination methods.
In equations “solvable for y,” the form typically is
A y = f(x,p)
B p = f(x,y)
C x = f(y,p)
D y = f(x) only
“Solvable for y” means y can be written in terms of x and p. Such forms are often treated using parametric methods and elimination of the parameter p.
A Clairaut equation has the standard form
A y′ + Py = Q
B y = px + f(p)
C y″ + ay = 0
D y = e^{ax}
Clairaut’s form is y = px + f(p), where p=dy/dx. Its general solution is a family of straight lines, and it may have a singular solution as an envelope.
The general solution of Clairaut’s equation is usually
A y = Ce^{x}
B y = C sin x
C y = Cx + f(C)
D y = C/x
In Clairaut’s equation, p becomes a constant C for the general family. Substituting p=C gives y = Cx + f(C), representing a one-parameter family of lines.
The singular solution of a Clairaut equation is found using
A Eliminating parameter
B Direct integration
C Laplace method
D Separation method
Differentiate y = px + f(p) treating p as variable, then use the relation from dy/dx and eliminate p. This produces the envelope curve, called the singular solution.
Geometrically, the singular solution of Clairaut’s equation represents the
A Intersection point
B Area under curve
C Period of motion
D Envelope of lines
The general solution gives a family of straight lines. The singular solution is the curve tangent to each member of the family, forming their envelope in the plane.
A typical Lagrange-type first-order equation has the form
A y = px + f(p)
B y′ + Py = Q
C y = xp + f(p)
D y″ + Py′ = 0
Lagrange’s form is often written y = xφ(p) + ψ(p). A common simple pattern is y = xp + f(p), handled using parameter methods and elimination.
The “discriminant method” in these equations is mainly used to find
A Singular solution
B Periodic solution
C Initial slope
D Integrating factor
In Clairaut/Lagrange-type equations, the singular solution can be located by treating the family as parametric and using envelope conditions, often connected to discriminant ideas.
For y = px + f(p), differentiating gives a relation involving
A dp/dx only
B (x + f′(p)) dp/dx
C y dp/dx
D p dx/dy
Differentiate y = px + f(p): dy/dx = p + x dp/dx + f′(p) dp/dx. Since dy/dx = p, we get (x + f′(p)) dp/dx = 0.
In Clairaut’s equation, the general solution represents
A Circles
B Parabolas
C Straight lines
D Hyperbolas
The general solution y = Cx + f(C) is a one-parameter family of straight lines. The singular solution, when present, is their common envelope curve.
A linear ODE with constant coefficients usually uses the
A Exactness test
B Bernoulli step
C Laplace only
D Auxiliary equation
For linear constant-coefficient ODEs, assume y=e^{mx}. Substitution yields the characteristic (auxiliary) polynomial in m. Its roots determine the complementary function form.
Distinct real roots of the characteristic equation produce
A Exponential CF terms
B Trigonometric CF
C Polynomial CF only
D Logarithmic CF only
If the characteristic equation has distinct real roots m₁, m₂, …, then the complementary function is a linear combination of e^{m₁x}, e^{m₂x}, and so on.
Repeated real root m of multiplicity 2 gives CF terms
A e^{mx}, e^{-mx}
B cos mx, sin mx
C e^{mx}, xe^{mx}
D 1, x only
For a repeated root m of multiplicity 2, the independent solutions are e^{mx} and x e^{mx}. Higher multiplicity adds x²e^{mx}, etc., ensuring independence.
Complex roots α ± iβ give CF terms
A e^{αx}cosβx, e^{αx}sinβx
B cosαx, sinαx
C e^{βx}cosαx only
D αx + β only
Complex conjugate roots lead to oscillatory solutions multiplied by an exponential factor. The real CF becomes e^{αx}(C₁cosβx + C₂sinβx).
“Complementary function” refers to solution of the
A Nonhomogeneous equation
B Homogeneous equation
C Exact equation only
D Separable equation
For linear ODEs, the complementary function (CF) solves the associated homogeneous equation. The complete solution is CF plus a particular integral for the nonhomogeneous part.
In constant coefficient ODEs, the “resonance case” occurs when
A RHS is zero
B Roots are complex
C x is missing
D RHS matches CF term
If the forcing term has the same form as a term in the complementary function, the trial particular solution must be multiplied by x (or higher powers) to remain independent.
The method of undetermined coefficients is mainly used when RHS is
A Standard simple forms
B Any arbitrary function
C Only discontinuous
D Only implicit
This method works well when RHS is a combination of polynomials, exponentials, sines, and cosines (and their products). Then a suitable trial form is chosen and coefficients solved.
In operator notation, D usually represents
A Integration
B Multiplication
C Differentiation
D Substitution
The operator D is shorthand for d/dx. Linear ODEs with constant coefficients can be expressed as polynomial operators in D applied to y, simplifying certain solution steps.
For Cauchy–Euler equations, a common trial solution is
A y = x^{m}
B y = e^{mx}
C y = sin mx
D y = ln x
Cauchy–Euler (equidimensional) equations have variable coefficients in powers of x. Trying y=x^{m} converts the equation into an algebraic equation in m.
A standard Cauchy–Euler equation is typically written using
A Constant coefficients only
B Trigonometric coefficients
C Random functions
D Powers of x coefficients
Cauchy–Euler equations have terms like x²y″, xy′, and y. Because coefficients follow powers of x, substitutions or power trials reduce them to solvable algebraic forms.
The substitution x = e^{t} is useful mainly to
A Make equation exact
B Remove initial conditions
C Convert to constant coefficients
D Force separable form
For Cauchy–Euler equations, setting x=e^{t} transforms derivatives with respect to x into derivatives in t, producing a linear ODE with constant coefficients in t.
“Reduction of order” is used when
A One solution is known
B No solution exists
C RHS is polynomial
D Equation is exact
For a second-order linear ODE, if one nonzero solution y₁ is known, reduction of order helps find a second independent solution y₂, completing the fundamental set.
Variation of parameters is mainly a method to find
A Complementary function
B Particular solution
C Characteristic roots
D Wronskian equals zero
Variation of parameters finds a particular solution of nonhomogeneous linear ODEs by allowing constants in the homogeneous solution to vary as functions, typically using the Wronskian.
For linear ODEs, the general solution commonly equals
A CF + PI
B CF − PI
C PI only
D CF only always
In a linear nonhomogeneous ODE, the complete solution is the sum of the complementary function (homogeneous solution) and a particular integral that accounts for the forcing term.
A “singular solution” is best described as
A Any constant solution
B Not in general family
C Always exponential
D Always periodic
A singular solution cannot be obtained by choosing a parameter value in the general solution family. It often arises as an envelope of the family, especially in Clairaut-type equations.
The “envelope” idea is closely connected to
A Boundary values
B Separable equations
C Singular solutions
D Laplace tables
The envelope of a one-parameter family of curves is tangent to each curve in the family. In differential equations like Clairaut’s, this envelope corresponds to the singular solution.
In linear ODEs, “homogeneous” means
A RHS is zero
B Coefficients are zero
C y is constant
D x is missing
A linear ODE is homogeneous if the forcing term is zero, so the equation equals 0. Solutions then form a vector space, enabling superposition and fundamental sets.
A “nonhomogeneous” linear ODE means
A No derivatives appear
B RHS is not zero
C Roots are complex
D Coefficients constant only
Nonhomogeneous linear ODEs include an external forcing term on the right side. Their solutions consist of a homogeneous part (CF) plus one particular solution.
The Wronskian of y₁ and y₂ for second order is
A y₁y₂′ − y₂y₁′
B y₁y₂
C y₁′y₂′
D y₁′ + y₂′
For two functions, the Wronskian is the 2×2 determinant formed by the functions and their first derivatives, giving W = y₁y₂′ − y₂y₁′.
If y₁ and y₂ are independent solutions, the general homogeneous solution is
A y₁y₂
B C₁y₁ + C₂y₂
C y₁ + y₂ only
D C(y₁ − y₂) only
In a second-order linear homogeneous ODE, any solution can be expressed as a linear combination of two independent solutions using constants C₁ and C₂.
For constant coefficient ODEs, assuming y=e^{mx} mainly converts the ODE into
A A trigonometric identity
B An integral equation
C A parametric curve
D An algebraic equation
Substituting y=e^{mx} turns derivatives into powers of m times e^{mx}. Cancelling e^{mx} yields the characteristic polynomial in m, solved algebraically.
A particular integral is required only when the equation is
A Nonhomogeneous
B Homogeneous
C Separable
D Exact only
If the RHS is not zero, the solution needs an additional part beyond the homogeneous solution. That additional part is the particular integral, matching the forcing term.
In undetermined coefficients, a polynomial RHS usually suggests a PI trial that is
A Exponential only
B Logarithmic only
C Polynomial type
D Trigonometric only
For polynomial forcing, try a polynomial of the same degree as the RHS. Adjust by multiplying by x if resonance occurs with the complementary function.
For RHS e^{ax}, the PI trial generally involves
A e^{ax}
B x^{a}
C sin ax
D ln x
Exponential forcing e^{ax} suggests a particular solution proportional to e^{ax}. If e^{ax} already appears in CF due to a root, multiply the trial by x enough times.
For RHS sin(ax) or cos(ax), PI trial usually uses
A Polynomial only
B Logarithmic terms
C Constant only
D A sin-cos pair
Because derivatives of sin and cos cycle, the trial must include both sin(ax) and cos(ax) terms. Coefficients are solved after substitution into the ODE.
A repeated root of multiplicity 3 leads to CF terms
A e^{mx}, e^{-mx}, xe^{mx}
B e^{mx}, xe^{mx}, x²e^{mx}
C cos mx, sin mx, e^{mx}
D 1, x, x² only
For a triple repeated root m, the independent solutions are e^{mx}, x e^{mx}, and x² e^{mx}. These maintain linear independence in the solution space.
A linear ODE with variable coefficients is “Cauchy–Euler” when coefficients are
A Powers of x
B Random functions
C Pure constants
D Trig functions
In Cauchy–Euler equations, coefficients scale with powers of x, such as x²y″ and xy′. This structure allows reduction using x^{m} trials or the substitution x=e^{t}.
“Operator D-method” mainly applies to
A Variable coefficient ODEs
B Only Clairaut equations
C Constant coefficient ODEs
D Only separable forms
The D-operator method treats derivatives as algebraic operators, which works smoothly for constant coefficient linear ODEs. It helps in forming CF and handling certain RHS forms.
Annihilator method is used to help find
A Particular solution
B Wronskian only
C Singular solution only
D Boundary condition
The annihilator method applies an operator that “kills” the RHS, converting the equation into a higher-order homogeneous one. Then the PI is inferred while avoiding duplication with CF terms.
A boundary value problem typically specifies
A y and y′ at same point
B Only slope at origin
C Only general solution
D Values at two points
Boundary value problems set conditions at different points, like y(a) and y(b). This differs from initial value problems, which specify conditions at the same starting point.
“Stability of solution” usually concerns
A Exactness only
B Root multiplicity only
C Behavior under perturbation
D Integral tables
Stability studies whether small changes in initial conditions cause small changes in solutions over time. It is an important qualitative idea, especially in modeling and applications.
In a linear ODE, if one particular solution is found, any other PI differs by
A A constant only
B A homogeneous solution
C A singular curve
D A Wronskian value
If y_p is one particular solution, then y_p + y_h is also a particular solution of the nonhomogeneous equation only when y_h solves the homogeneous equation, explaining the “difference” rule.