Second-Order Linear Equations

Calculus 2 · Axiom Academy

LESSON Second-Order Linear Equations Guess y=e^ rt , collapse the equation to a quadratic, and let its discriminant decide the shape of the solution. 1. One Guess Turns It Into a Quadratic Every term of ay''+by'+cy=0 is a derivative of y , and the one function whose derivatives are all copies of itself is the exponential. So we guess y=e^ rt . Then y'=re^ rt and y''=r^2e^ rt , and after factoring out the never-zero e^ rt , the differential equation becomes a plain quadratic in r . Substitute the guess and its derivatives Cancel → the characteristic equation 2. The Discriminant Decides Everything The characteristic equation is a quadratic, so the discriminant tells us how many real roots it has — and therefore which solution form we land in. Watch a single parabola lift its vertex through the r -axis: the number of places it crosses goes from two, to one, to none. Each kind of root builds the general solution from two independent pieces . Distinct real roots give two separate exponentials. A repeated root reuses e^ rt and multiplies the second copy by t . A complex pair, unwound with Euler's formula, becomes an exponential times a sine and cosine. Three forms, three unmistakable shapes. Example. y''-3y'+2y=0 factors as (r-1)(r-2)=0 , so and y=c_1e^ t +c_2e^ 2t . Example. y''-4y'+4y=0 gives (r-2)^2=0 , a repeated r=2 , so y=(c_1+c_2t)e^ 2t . Example. y''+2y'+5y=0 gives , so and .

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