MZB127 Chap.5 Second-Order Differential Equations
Second-Order Differential Equations
Constant-coefficient second-order equations reduce to a characteristic polynomial for the homogeneous motion. Distinct real, repeated and complex roots create different two-constant solution families. Nonhomogeneous forcing adds a particular solution, with resonance requiring a modified trial.
The chapter connects root signs to growth, decay and oscillation, then applies both initial conditions and substitutes the complete result back into the equation. The two independent basis functions in the complementary solution reflect the two pieces of initial data a second-order model normally needs.
Distinct real roots yield two exponential modes, a repeated root requires a time-multiplied second mode, and complex conjugates become a real sine-cosine pair under an exponential envelope. Forcing introduces a steady or driven component whose trial shape follows the right-hand side. If that trial overlaps the complementary family, multiplication by an adequate power of time restores independence.
The real parts of roots predict growth or decay, while imaginary parts set oscillation. These qualitative features can be forecast before constants are calculated and checked afterward. A complete answer combines complementary and particular parts, uses both initial conditions and verifies that applying the differential operator reproduces the forcing.
What this chapter covers
- 01
Second-order linear classification
- 02
Characteristic polynomial and roots
- 03
Distinct, repeated and complex solution forms
- 04
Undetermined coefficients
- 05
Resonance and initial conditions
Worked example · free
Recover damped oscillation
- 1Solve the characteristic equation.
- 1Write the complex-root solution family.
- 1Apply the position condition.
- 1Differentiate and apply the velocity condition.
- 1Interpret decay and oscillation from the roots.
Key terms
- Characteristic equation
- The polynomial equation obtained from an exponential trial in a homogeneous constant-coefficient ODE.
- Complementary solution
- The complete homogeneous solution family associated with the characteristic roots.
- Particular solution
- One nonhomogeneous solution that reproduces the stated forcing.
- Resonance
- Overlap between a forcing trial and the homogeneous family, requiring an independent modified trial.
- Damped oscillation
- Oscillatory motion whose amplitude decreases through a negative exponential envelope.
Second-Order Differential Equations FAQ
Why does a repeated root introduce t?
One exponential supplies only one independent solution. Multiplying by t produces a second independent function, allowing the two constants needed to satisfy two initial conditions.
How is resonance detected?
Compare the proposed particular trial with the complementary solution. If the trial already appears in the homogeneous family, multiply the whole trial family by a sufficient power of t before coefficient matching.
What do complex roots mean physically?
Their imaginary part creates sine and cosine oscillation, while the real part creates the exponential envelope. Negative real part gives decay, positive real part gives growth, and zero produces undamped ideal oscillation.
How many constants should a second-order solution contain?
Before initial data are applied, a complete second-order solution normally carries two independent arbitrary constants. If only one appears, inspect the root case for a missing independent basis function. After forcing is included, the particular solution adds no new arbitrary constant because its coefficients are fixed by the differential equation.
Exam move
Write the differential equation, characteristic polynomial and root classification on separate lines. Memorise the three homogeneous forms by understanding why two independent functions are required. For forcing, write the complementary family before choosing the particular trial and test for overlap. Apply initial data only after combining homogeneous and particular parts.
Check the final function in both the equation and conditions. Drill the root cases by moving in both directions: derive a solution form from given roots, then infer the roots and stability from a displayed solution. Count two independent arbitrary constants before using initial data.
For complex roots, label the real part as the envelope rate and the imaginary magnitude as the oscillation rate, then predict the qualitative graph. On nonhomogeneous questions, place the complementary solution and forcing type in separate boxes. Propose the full trial family, compare every term with the homogeneous basis and multiply the entire overlapping family by time when resonance occurs.
Differentiate the trial carefully and display coefficient matching rather than jumping to fitted constants. Only after complementary and particular terms are combined should initial position and velocity be imposed. Differentiate the complete expression symbolically before setting time to zero so product-rule terms cannot disappear.
Verify the final result three ways: both initial conditions, substitution into the differential equation, and agreement between root signs and long-run behaviour. Maintain an error log for characteristic-sign mistakes, missing time factors, incomplete trigonometric pairs and premature initial substitution.
Timed practice should still show the polynomial, root classification, trial justification and verification, because those lines reveal method decisions even if later arithmetic slips.
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