First Order Odes When to Use Use this skill when working on first-order-odes problems in odes pdes. Decision Tree 1. Classify the ODE - Linear: y' + P(x)y = Q(x)? - Separable: y' = f(x)g(y)? - Exact: M(x,y)dx + N(x,y)dy = 0 with dM/dy = dN/dx? - Bernoulli: y' + P(x)y = Q(x)y^n? 2. Select Solution Method | Type | Method | |------|--------| | Separable | Separate and integrate | | Linear | Integrating factor e^{int P dx} | | Exact | Find potential function | | Bernoulli | Substitute v = y^{1-n} | 3. Numerical Solution (IVP) - - For stiff systems: or - Adaptive step size: specify rtol/atol, not…