Solving x' = [1,2;3,2]x

Differential Equations · Axiom Academy

EXAMPLE Solving with Eigenvalues Find the eigenvalues and eigenvectors of a system to build its general solution Solve the system of differential equations where Excellent work! You solved a system of differential equations by finding eigenvalues and eigenvectors, then combined them into a general solution. Characteristic equation: for , the eigenvalues solve . Eigenvectors: for each eigenvalue , solve to find its eigenvector (defined up to a scalar multiple). General solution: combine every eigenvalue-eigenvector pair, . Verification: always confirm a candidate solution by substituting back into and checking the residual is . Real eigenvalues: when they're real and distinct, the solution is a sum of growing/decaying exponentials along each eigenvector direction. Here and — real eigenvalues of opposite sign, so the origin is a saddle point : trajectories are pulled in along but pushed out along .

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