Day 6 — Differential Equations

Time: ~45 min · Topic: differential equations

The problem

Solve the differential equation y'' - 5y' + 6y = 2e^(3x) with initial conditions y(0) = 1 and y'(0) = 4.

Hint: One root of the characteristic equation matches the exponent on the right side — so your usual guess for the particular solution needs a twist.

Try it yourself

Take a few minutes. Don't peek at the steps. If you get stuck, the hint above is enough to nudge you forward.

Step-by-step solution

1. Solve the characteristic equation r^2 - 5r + 6 = 0. Factor as (r - 2)(r - 3) = 0, giving roots r = 2 and r = 3. The homogeneous solution is y_h = C1e^(2x) + C2e^(3x).
2. Find a particular solution. Since e^(3x) already appears in y_h (r = 3 is a root), we multiply by x and try y_p = Ax*e^(3x). Compute y_p' = A(1 + 3x)e^(3x) and y_p'' = A(6 + 9x)e^(3x).
3. Substitute into the ODE: A(6 + 9x)e^(3x) - 5A(1 + 3x)e^(3x) + 6Axe^(3x) = 2e^(3x). Simplify the left side: A(6 + 9x - 5 - 15x + 6x)e^(3x) = A(1)e^(3x). So A = 2, and y_p = 2xe^(3x).
4. The general solution is y = C1e^(2x) + C2e^(3x) + 2xe^(3x). Apply y(0) = 1: C1 + C2 = 1. Compute y' = 2C1e^(2x) + 3C2e^(3x) + 2e^(3x) + 6xe^(3x), then y'(0) = 4 gives 2C1 + 3C2 + 2 = 4, so 2C1 + 3C2 = 2. Solving: C2 = 0 and C1 = 1.

Answer: y = e^(2x) + 2x*e^(3x)

Why this matters

Differential equations like this one govern how shock absorbers in your car work — the equation models a damped spring being hit by an outside force!

This day in math: The Man Who Tamed Randomness

April 26, 1903 · Andrei Kolmogorov

On April 25, 1903, Andrei Kolmogorov was born in Tambov, Russia. He would go on to become one of the most influential mathematicians of the 20th century, revolutionizing probability theory with his 1933 monograph that laid down just three elegant axioms — giving the entire field its rigorous mathematical foundation for the first time. His work spanned an astonishing range: turbulence theory, algorithmic complexity (now called Kolmogorov complexity), topology, and classical mechanics.

Why it matters: Before Kolmogorov, probability was more intuition than mathematics. His axioms unified the field under measure theory, making modern statistics, machine learning, quantum mechanics, and financial modeling possible.

Did you know?

If you randomly pick two whole numbers, the probability that they share no common factor (other than 1) is exactly 6 / pi^2 — roughly 61%. Pi, a number born from circles, mysteriously appears in a problem that has absolutely nothing to do with geometry.

This result comes from the Euler product over all prime numbers. The chance that two random numbers are both NOT divisible by a given prime p is (1 - 1/p^2). Multiplying this across every prime gives 1/zeta(2), and the great mathematician Euler proved that zeta(2) = pi^2 / 6. So the final probability simplifies beautifully to 6 / pi^2, connecting prime numbers to circles in one of math's most unexpected bridges.

Done?

• Solved the problem (or read through the steps)
• Read the history note
• Read the "did you know" fact