Undetermined Coefficients Theory

Differential Equations · Axiom Academy

LESSON Undetermined Coefficients Theory Match the trial solution's form to the driving function's form, substitute, and solve for the coefficients — a systematic guessing game with one crucial exception. 1. The Big Idea: Let the Form of g(x) Suggest y p The general solution is y = y_h + y_p , where y_h solves the homogeneous equation ( g(x)=0 ) and y_p is any one solution that produces the actual g(x) on the right side. 2. The Systematic Trial-Form Table Here is the standard table matching each driving-function family to its trial particular solution (undetermined coefficients A , B , to be solved for): 3. Worked Example: A Clean (Non-Resonant) Case Solve for a particular solution of: 1. Characteristic roots of y_h : , so y_h=c_1e^ x +c_2e^ 2x . 2. Check for resonance: g(x)=e^ 3x has rate . Is 3 a characteristic root? No — the roots are 1 and 2 . No resonance , so the standard trial applies. 3. Trial (from the table): y_p = Ae^ 3x . 4. Differentiate and substitute: y_p'=3Ae^ 3x , y_p''=9Ae^ 3x , so 4. The Resonance Problem — When the Trial Fails Substituting a homogeneous solution into L[y]=ay''+by'+cy always gives zero — by definition, that's what "homogeneous solution" means. So if the naive trial is a homogeneous solution, L[y_p] collapses to 0 no matter what the coefficients are — it can never equal a nonzero g(x) . This clash is called resonance , and the standard trial from the table is guaranteed to fail. 5. Worked Example: Resonance in Action

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