Benjamin's Maths World

Second-order Constant-coefficient Homogeneous ODEs, Auxiliary Equation: A Repeated Real Roots.

Second-order Constant-coefficient Homogeneous ODEs, Auxiliary Equation: Two Real Roots.

Focusing on the case of second-order linear ODEs. Wronskian.

Picard–Lindelöf Theorem: Lipschitz Continuity, Picard’s successive iterations.

Orthogonal trajectories by replacing \( dy/dx \) by \( -dx/dy \).

First-order higher-degree ODEs soluble for \( y \) or \( x \).

For a higher-degree first-order ODE in the form of a polynomial in \( y’ \).

Riccati ODEs are the mixture of first-order linear ODEs and Bernoulli ODEs.

Bernoulli ODEs: \(dy/dx + P(x)y = Q(x)y^n\), by substituting \( v = y^{1-n}\).

Isobaric ODEs are the generalization of homogeneous ODEs discussed in the last tutorial.