Guessing Particular Solutions

Differential Equations · Axiom Academy

Discover the pattern-matching trick behind undetermined coefficients — the form of your guess mirrors the driving term on the right-hand side. Let the Right-Hand Side Guide Your Guess To solve a nonhomogeneous equation like y''+by'+cy=g(x) , you need one particular solution y_p that matches g(x) . Rather than searching blindly, you can guess the FORM of y_p just by looking at g(x) — because differentiation keeps a function inside its own family. Watch the trial solution y_p=Ae^ 4x get substituted into y''-3y'+2y=e^ 4x : each derivative of e^ 4x is still a multiple of e^ 4x , so the whole left side collapses into one exponential term — leaving a simple equation to solve for A . Every derivative of e^ 4x is just a constant times e^ 4x again — that's WHY an exponential driving term always pairs with an exponential guess. Match Each Driving Term To Its Guess Form Drag each driving term on the left to the guess form it belongs with on the right. Think about what happens when you differentiate each type of function. Each family is closed under differentiation: derivatives of e^ kx stay exponential, derivatives of and stay in that same pair, and derivatives of a degree- n polynomial stay polynomial (just lower degree) — so the guess must include every term down to a constant. What If the Driving Term Is a Sum? Click the guess you think matches a driving term that combines TWO families at once.

This is the written version of the interactive lesson above. See the full Differential Equations course.