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Mathematics · lesson 2 of 3

Second-order equations and resonance

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

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Learn the ideas

Prerequisites: Complex roots and elementary differential equations.

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.

Worked examples

Example 1

Solve y″+4y=0 with y(0)=3,y′(0)=0.

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

y(t)=3cos(2t).

Example 2

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

r²−3r+2=(r−1)(r−2).

2

Check your understanding

8 questions

Decimals or fractions are accepted for numeric answers. Proofs and written responses use sample answers and require human marking. This is practice, not a protected examination.

1 Solve y″+4y=0 with y(0)=3,y′(0)=0.

Sample answer and marking guidance

y(t)=3cos(2t).

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

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

Show answer and explanation

2

r²−3r+2=(r−1)(r−2).

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

Show answer and explanation

e^t and te^t

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 and marking guidance

A suitable trial is A t sin t; substitution gives A=1/2, so y_p=(t/2)sin t.

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

5 For y(t)=2cos(3t), calculate y″(0).

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-18

Differentiating twice gives −18cos(3t).

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

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y(t₀) and y′(t₀)

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 and marking guidance

It is a scalar multiple, so their span is one-dimensional. Two constants multiplying these terms can be combined into one.

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?

Show answer and explanation

An amplitude envelope growing with time

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

Further reading and review references

These are reference links, not copied exercises, institutional endorsement or proof of independent review.