Problem-Solving: Differential Equations

Differential equations and vector calculus problems require choosing the right technique for each situation.

ODE decision tree

First-order? Check if separable, linear, exact, or Bernoulli.
Second-order with constant coefficients? Write the characteristic equation.
Non-homogeneous? Find $y_h$ first, then $y_p$ by undetermined coefficients or variation of parameters.
System? Write as $\vec{x}' = A\vec{x}$ and find eigenvalues.
Initial-value problem with discontinuities? Use Laplace transforms.

Vector calculus decision tree

Need to compute	Tool
Work along a path	Line integral $\int\vec{F}\cdot d\vec{r}$
Flux through surface	Surface integral $\iint\vec{F}\cdot d\vec{S}$
Simplify closed-surface flux	Divergence theorem
Simplify circulation integral	Stokes' theorem
Check if $\vec{F}$ is conservative	$\nabla\times\vec{F} = \vec{0}$ ?

from sympy import symbols, Function, dsolve, Eq
t = symbols('t')
y = Function('y')
ode = Eq(y(t).diff(t, 2) + 4*y(t), 0)
sol = dsolve(ode, y(t))
print(sol)

Verification checklist

Substitute solution back into the original ODE.
Check that initial/boundary conditions are satisfied.
Confirm units are consistent throughout.

Tip: The hardest part of DEs is recognising the type. Once classified, the solution method is mechanical. Practice classification on sight.

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 sides look equally hard, you have not yet exploited the freedom to choose the surface or region.

Practise this lesson

Differential Equations and Vector Calculus