Determine the roots of the characteristic equation for the following homogeneous differential equation

1

2
d2x

dt2
−7
dx

dt
+3x=0

One root is greater than the other (i.e if you placed the two roots on a number line, the greatest root would be the one to the right of the other).

Enter the greatest root below, using 2 decimal places.

Hint: you may need to use the quadratic formula.

Question

Determine the roots of the characteristic equation for the following homogeneous differential equation

1

2
d2x

dt2
−7
dx

dt
+3x=0

One root is greater than the other (i.e if you placed the two roots on a number line, the greatest root would be the one to the right of the other).

Enter the greatest root below, using 2 decimal places.

Hint: you may need to use the quadratic formula.

BlackTom AI · Accepted Answer

For the ODE (1/2) x'' − 7 x' + 3 x = 0 the characteristic polynomial is (1/2) r^2 − 7 r + 3 = 0.

1) Relevant knowledge:
- For a second-order linear homogeneous ODE with constant coefficients a x'' + b x' + c x = 0, the characteristic equation is a r^2 + b r + c = 0.
- Solve that quadratic with the quadratic formula r = [−b ± sqrt(b^2 − 4ac)]/(2a).

2) Step-by-step solution:
Start with (1/2) r^2 − 7 r + 3 = 0.
Multiply both sides by 2 to clear the fraction:
r^2 − 14 r + 6 = 0.

Apply the quadratic formula with a = 1, b = −14, c = 6 (or directly treat as r^2 − 14r + 6):
Discriminant D = b^2 − 4ac = (−14)^2 − 4(1)(6) = 196 − 24 = 172.

Compute sqrt(D):
sqrt(172) = sqrt(4·43) = 2 sqrt(43) ≈ 13.114877.

Now find the two roots:
r = [14 ± sqrt(172)] / 2.

Largest root:
r_max = (14 + 13.114877) / 2 = 27.114877 / 2 = 13.5574385 ≈ 13.56 (rounded to two decimals).

Smaller root:
r_min = (14 − 13.114877) / 2 = 0.885123 / 2 = 0.4425615 ≈ 0.44.

3) Intermediate steps shown above include clearing the fraction, computing discriminant, sqrt, and final division.

4) Why this is correct:
Multiplying by 2 is necessary because the leading coefficient is 1/2. Solving the correctly formed quadratic yields the two real roots shown; the larger of these two is ≈13.5574, which rounds to 13.56.

5) Common incorrect approaches:
- If one mistakenly forms r^2 − 7r + 3 = 0 (i.e., forgetting the 1/2 factor), the larger root would be (7 + sqrt(37))/2 ≈ 6.54, which is incorrect for the original equation.
- Arithmetic or rounding errors in sqrt(172) can also give wrong final rounding.

Conclusion: The greatest root, rounded to two decimal places, is 13.56, matching the provided answer.

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