21/3848–12FE Reference Handbook 10.6 · Mathematics · p. 54 Second-order linear homogeneous DE
1. Quadratic Formula2. Law of Cosines3. Standard Normal z-Score4. Law of Sines5. Power Rule6. Definite Integral of a Power7. Dot (Scalar) Product8. Cross Product Magnitude9. Sample Standard Deviation10. Binomial Probability11. z Confidence Interval for the Mean12. Circle: Area, Circumference, Arc13. First-Order Linear DE14. Trapezoidal Rule15. Linear Least Squares16. Poisson Probability Mass Function17. Newton–Raphson Iteration18. 2×2 Determinant and Inverse19. Combinations C(n,k)20. Linear Interpolation21. Second-Order Linear DE (Constant Coeff.)22. Exponential Growth and Decay23. Straight Line24. Mean, Median, and Mode25. Addition Rule and Independence26. Derivative of ln and e^u27. Vector Length and Direction Cosines

Handbook formula

Second-Order Linear DE (Constant Coeff.)

p. 54 writes the monic equation y'' + a y' + b y = 0. Roots are real and distinct if a² > 4b, real and equal if a² = 4b, and complex if a² < 4b. Do not put the y' coefficient into a 2a denominator — that denominator is for a non-monic polynomial. Divide by the leading coefficient first. Fourier transforms on the same page are outside the FE Civil math specification.

Monic coefficients of y' and y
Characteristic root

Step-by-step solved example

y''+5y'+6y=0. Find the general solution.

e^(−2t)e^(−3t)Replay
Free oscillator: the same time axis as kinematics, now as y(t).

  1. 1. Roots

    Monic a = 5, b = 6. r = [−5 ± √(25 − 24)] / 2 = −3 and −2.

  2. 2. Solution

    y=C1 e^{−2t}+C2 e^{−3t}.

Answer: y=C1 e^{−2t}+C2 e^{−3t}

10 practice questions

0/10 correct

1.Repeated root r=−4 gives y =

2.y''+ω²y=0 solutions are

3.For y''+ay'+by=0, a² < 4b means

4.Particular solution of y''+y=2 is

5.y''−y=0 roots are

6.Initial y(0), y'(0) fix

7.A first-order system equivalent uses

8.Overdamped free motion

9.y''+2y'+y=0 is

10.Forcing sinΩt on an undamped oscillator at Ω=ω causes