Second-Order Linear Equations with Constant Coefficients: every key term you need (+ practice quiz)
25 flashcard terms for Differential Equations Topic 4, written to match the course framework. Study them here, then drill them as interactive flashcards, or test yourself with the 15-question quiz — free, no account needed.
An equation in which the second derivative, first derivative and unknown appear linearly with coefficient functions of the independent variable, plus a forcing term. Its general solution needs two arbitrary constants.
Homogeneous equation
A linear equation whose forcing term is identically zero. Its solutions form a vector space, so any linear combination of solutions is again a solution, which is the structural fact the whole theory rests on.
Superposition principle
For a homogeneous linear equation, sums and scalar multiples of solutions are solutions. It fails immediately for nonlinear equations, which is why linear problems are so much more tractable.
Fundamental set of solutions
Two solutions of a second-order homogeneous equation whose linear combinations generate every solution. Existence of such a pair on an interval follows from the existence and uniqueness theory for linear equations.
Wronskian
The determinant formed from two solutions and their first derivatives. It is nonzero somewhere on an interval exactly when the pair is linearly independent there and therefore forms a fundamental set.
Abel identity
The statement that the Wronskian of two solutions satisfies a first-order equation driven by the coefficient of the first derivative, so it is either identically zero or never zero on the interval.
Characteristic equation
The quadratic obtained by substituting an exponential trial solution into a constant-coefficient homogeneous equation. Its roots determine the exponential rates present in the general solution.
Distinct real roots case
When the characteristic quadratic has two different real roots, the general solution is a combination of the two corresponding real exponentials. Solutions cross zero at most once and are dominated by the larger root at long times.
Repeated root case
When the characteristic quadratic has one root twice, the exponential is one solution and the independent variable times that exponential is a second, independent one. Reduction of order explains where the extra factor comes from.
Complex conjugate roots case
When the roots are complex, the real general solution is an exponential in the real part multiplying a combination of cosine and sine of the imaginary part. Complex exponentials are converted using the Euler formula.
Euler formula for solutions
The identity expressing a complex exponential as cosine plus the imaginary unit times sine. It is what turns the complex root pair into a pair of real oscillatory solutions with a real exponential envelope.
Reduction of order
A technique that starts from one known solution and seeks a second as that solution multiplied by an unknown function. Substituting reduces the problem to a first-order equation for the derivative of the unknown factor.
Linear independence of functions
Two functions are independent on an interval if neither is a constant multiple of the other there. For solutions of a second-order linear equation this is detected by a nonvanishing Wronskian.
General solution structure for forced equations
The general solution of a nonhomogeneous linear equation is any one particular solution plus the general solution of the associated homogeneous equation. Both halves must be found before initial conditions are imposed.
Complementary solution
The general solution of the homogeneous version of a forced equation. It carries the two arbitrary constants and, for stable equations, is the transient part that dies away.
Boundary value problem
A second-order equation with conditions imposed at two different points rather than at one. Unlike an initial value problem, it may have no solution or infinitely many, depending on the equation and the interval.
Existence and uniqueness for linear second-order equations
If the normalised coefficients and forcing are continuous on an interval containing the initial point, a unique solution exists on that entire interval for any pair of initial values.
Operator notation
Writing the left side of a linear equation as a differential operator applied to the unknown. The operator is linear, and factoring it into first-order operators mirrors factoring the characteristic polynomial.
Cauchy Euler equation
A variable-coefficient equation whose coefficients are powers of the independent variable matching the derivative order. A power trial solution reduces it to an auxiliary polynomial, so it is solvable much like a constant-coefficient equation.
Higher-order constant-coefficient extension
The same characteristic polynomial approach works for equations of any order, with a root of multiplicity m contributing that exponential multiplied by successive powers up to m minus one.
Sign of the roots and long-time behaviour
All solutions of a constant-coefficient homogeneous equation tend to zero exactly when every characteristic root has negative real part. A single root with positive real part makes almost every solution grow without bound.
Discriminant of the characteristic quadratic
The quantity distinguishing the three solution cases. Positive gives distinct real exponentials, zero gives the repeated root form, and negative gives an oscillatory solution with exponential envelope.
Initial conditions for second-order problems
Two numbers are needed, typically the value and the derivative at one point. Substituting the general solution and its derivative gives a two by two linear system for the arbitrary constants.
Zeros of oscillatory solutions
When the roots are complex, every nontrivial solution has infinitely many zeros spaced by a fixed half period determined by the imaginary part, no matter what the arbitrary constants are.
Solution space dimension
The solution set of a homogeneous linear equation of order n is a vector space of dimension exactly n. This is why exactly n independent solutions are needed and why any extra solution must be a combination of them.