Constant-coefficient characteristic equation
Turn a homogeneous linear differential equation into an algebraic root problem.
Constant-coefficient characteristic equation is one of 2 second order formulas in the differential equations section of this library, and it is used at university level.
Why constant-coefficient characteristic equation works
Guess a solution of the form e^(rx). Each derivative pulls out another factor of r while leaving the exponential untouched, so the whole equation becomes e^(rx) times a quadratic in r. An exponential is never zero, so it divides out and the differential equation collapses into ordinary algebra.
What each symbol means
$r$ is the characteristic root.
Constant-coefficient characteristic equation: when it holds
$a\ne0$; solution form depends on distinct, repeated, or complex roots.
When it stops applying
The exponential guess only pays off when the coefficients are constants. An equation like x²y'' + xy' − y = 0 has coefficients that vary with x, so e^(rx) never simplifies away; that Euler type needs the guess y = x^r instead.
Constant-coefficient characteristic equation: a worked example
$y''-3y'+2y=0$ has roots $1,2$, so $y=C_1e^x+C_2e^{2x}$.
The mistake to avoid
What people do: Students solve for the roots and then present those numbers as the answer, writing y = C₁(1) + C₂(2) for roots 1 and 2.
Why it goes wrong: The roots are exponents, not the solution. Dropping the exponential loses the function entirely and leaves a constant that cannot satisfy a differential equation.
Do this instead: Put each root back into an exponential. For y'' − 3y' + 2y = 0 the roots 1 and 2 give y = C₁e^x + C₂e^(2x), which you can verify by differentiating twice.
Constant-coefficient characteristic equation: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $r$ is the characteristic root.
- Check the conditions before substituting. $a\ne0$; solution form depends on distinct, repeated, or complex roots.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most differential equations slips.
Where this formula fits
- Subject
- Differential Equations formulas — 8 entries in this library
- Topic
- Second order
- Level
- University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where constant-coefficient characteristic equation comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Integration Techniques — the lesson behind this formula: choose substitution, parts, or simplification from structure.
- Calculus Calculator — check your substitution and the value it produces.
- Study differential equations — the subject guide that explains the ideas these formulas compress.
- Differential Equations Practice — questions that make you retrieve the formula instead of recognising it.
- All 8 differential equations formulas — the full grouped reference, or the complete formula library.
Questions about constant-coefficient characteristic equation
Why guess an exponential in the first place?
Because the exponential is the one function whose derivative is a multiple of itself. That property is exactly what lets every term collapse into a common factor.
What if the roots come out complex?
That signals oscillation. A pair α ± iβ turns into e^(αx) times a combination of cos βx and sin βx, so the real part controls growth or decay and the imaginary part sets the frequency.
Where does the right-hand side go if it is not zero?
It gets handled separately. Solve the homogeneous version first for the complementary solution, then find any one particular solution for the forcing term and add the two together.
Does this extend to third-order equations?
Yes. A third-order constant-coefficient equation gives a cubic in r, and each root contributes its own exponential, with the same adjustments for repeats and complex pairs.