# Second-order equations and resonance

> For constant coefficients, an exponential trial function reduces the homogeneous ODE to a polynomial.

Course: [Differential equations and numerical methods](https://studybrix.com/en/courses/differential-equations/index.md)
Level: university
Prerequisites: Complex roots and elementary differential equations.

## Notes

### Characteristic equation

For constant coefficients, an exponential trial function reduces the homogeneous ODE to a polynomial.

### Initial data

A second-order equation usually requires two initial conditions for a unique local solution under standard hypotheses.

### Repeated roots

A repeated root r needs both e^(rt) and te^(rt).

### Resonance

A forcing term that duplicates a homogeneous solution requires an extra power of t in a particular-solution trial.

## Practice questions

### 1. Solve y″+4y=0 with y(0)=3,y′(0)=0.

Sample answer / marking guidance: y(t)=3cos(2t).

Explanation: The characteristic roots are ±2i, and the initial conditions remove the sine term.

### 2. For y″−3y′+2y=0, give the larger characteristic root.

Answer: 2

Explanation: r²−3r+2=(r−1)(r−2).

### 3. What pair spans the solutions of y″−2y′+y=0?

A. sin t and cos t
B. e^t and te^t
C. 1 and t²
D. e^t and 2e^t

Answer: e^t and te^t

Explanation: The repeated root r=1 needs the generalized solution te^t.

### 4. Give a valid form for a particular solution of y″+y=cos t.

Sample answer / marking guidance: A suitable trial is A t sin t; substitution gives A=1/2, so y_p=(t/2)sin t.

Explanation: Credit the resonance adjustment and substitution, not an ordinary A cos t trial alone.

### 5. For y(t)=2cos(3t), calculate y″(0).

Answer: -18

Explanation: Differentiating twice gives −18cos(3t).

### 6. Which initial data are sufficient for a standard second-order IVP?

A. Three arbitrary values of y at the same point
B. y(t₀) and y′(t₀)
C. Only y(t₀) for every equation
D. Only an equation title

Answer: y(t₀) and y′(t₀)

Explanation: Two independent initial conditions determine the two constants under the usual existence hypotheses.

### 7. Explain why 2e^t does not add an independent solution direction to e^t.

Sample answer / marking guidance: It is a scalar multiple, so their span is one-dimensional. Two constants multiplying these terms can be combined into one.

Explanation: Credit linear dependence and the missing second independent solution.

### 8. In an undamped forced oscillator at its natural frequency, what does the ideal resonant model permit?

A. An amplitude envelope growing with time
B. Instantaneous loss of all energy
C. No response
D. A guaranteed constant amplitude

Answer: An amplitude envelope growing with time

Explanation: Ideal resonance transfers energy coherently; real damping limits the idealized growth.

## Further reading

- https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/
- https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/

Original AI-assisted educational draft; not independently teacher-reviewed. References support further study, not copied exercises, affiliation or curriculum accreditation.