Problem-Solving: Differential Equations

ODE Problem-Solving

Differential equations and vector calculus problems require choosing the right technique for each situation. The hardest part of DEs is recognising the type; once classified, the solution method is mechanical.

ODE decision tree

Problem Type	Action / Technique
First-order	Check if separable, linear, exact, or Bernoulli
Constant coefficients	Write the characteristic equation
Non-homogeneous	Find $y_h$ first, then $y_p$ via undetermined coefficients
Systems	Write as $\vec{x}' = A\vec{x}$ and find eigenvalues
Discontinuities	Use Laplace transforms

Verification checklist: Substitute the solution back into the original ODE, check initial or boundary conditions, and confirm consistent units.

Three equations, three branches, one tree

Vector Calculus Decisions

Vector calculus requires picking the correct integral tool for the physical quantity you need to compute.

Vector calculus decision tree

Need to compute	Tool
Work along path	Line integral $\int\vec{F}\cdot d\vec{r}$
Flux through surface	Surface integral $\iint\vec{F}\cdot d\vec{S}$
Closed-surface flux	Divergence theorem
Circulation integral	Stokes' theorem
Conservative test	Check if $\nabla\times\vec{F} = \vec{0}$

Common pitfall: The big theorems (Green, Stokes, divergence) trade a boundary integral for an interior one. The win comes from picking whichever side is easier. If both look equally hard, you missed a simpler choice.

ODE Problem-Solving

Vector Calculus Decisions

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Differential Equations and Vector Calculus