Maths › Differential equations › Second order equations
Second order equations
Guess e to the mx, and a differential equation collapses to a quadratic: its two roots, real, repeated or complex, dictate everything the solution can do.
Builds on First order equations and integrating factors and Complex arithmetic and the Argand diagram.
IN THIS TOPIC
- Reduce ay'' + by' + cy = 0 to its auxiliary quadratic and classify the roots.
- Write the general solution in each of the three root cases.
- Use two initial conditions to pin both constants.
WHAT YOU PROBABLY THINK
A differential equation with a second derivative in it is a whole order of magnitude harder than a quadratic.
The auxiliary quadratic
Try y = e^(mx) in ay'' + by' + cy = 0: every term carries e^(mx), which cancels, leaving the auxiliary equation:
The differential equation is exactly as hard as this quadratic, no more, and the opening claim collapses with it. Distinct real roots m₁ and m₂ give y = Ae^(m₁x) + Be^(m₂x); a repeated root m gives y = (A + Bx)e^(mx); complex roots p ± qi give y = e^(px)(A cos qx + B sin qx), an oscillation inside an exponential.
WORKED EXAMPLE
Distinct roots with initial conditions
Solve y'' − 5y' + 6y = 0 with y(0) = 0 and y'(0) = 1.
Auxiliary: m² − 5m + 6 = (m − 2)(m − 3) = 0, so m = 2, 3.
General solution: y = Ae2x + Be3x.
y(0) = 0: A + B = 0. y'(0) = 1: 2A + 3B = 1. So B = 1, A = −1.
y = e3x − e2x: two conditions, two constants, both used.
When the roots turn complex
Complex roots do not signal a mistake; they signal oscillation. For y'' + 4y = 0 the auxiliary m² + 4 = 0 gives m = ±2i, so y = A cos 2x + B sin 2x: pure oscillation at angular frequency 2, the differential equation behind simple harmonic motion. The real part of the root controls growth or decay; the imaginary part sets the frequency.
YOUR TURN
A repeated root
Solve y'' − 4y' + 4y = 0.
Show the working
Auxiliary: m² − 4m + 4 = (m − 2)² = 0, repeated root m = 2.
The repeated case needs its extra x: y = (A + Bx)e2x.
Without the Bx term the 'two' solutions would be one in a wig, and two initial conditions could not both be met.
THE EXAM BIT
- Write the auxiliary equation down as a labelled step; the method mark attaches to it.
- The repeated-root case must carry (A + Bx); forgetting the x is the classic error.
- For complex roots p ± qi, e^(px) takes the real part and cos/sin take the imaginary part.
- Differentiate the general solution before substituting a y' condition, not after.
CHECK YOURSELF
Find the roots of the auxiliary equation for y'' + y' − 6y = 0 and write the general solution.
Show a hint
m² + m − 6 factorises.
Show the answer
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Substitute e^(mx): the equation becomes am² + bm + c = 0, and the roots classify the motion.
Real roots decay or grow; repeated roots need (A + Bx); complex p ± qi oscillate at frequency q inside e^(px).
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
CHECK YOUR PROGRESS
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- Reduce ay'' + by' + cy = 0 to its auxiliary quadratic and classify the roots.
- Write the general solution in each of the three root cases.
- Use two initial conditions to pin both constants.
Open the full revision checklist to see every objective in the course in one place.
No animated video for this topic yet; these notes stand alone.