What We Cover
First-Order ODEs
- Separable equations
- Linear first-order (integrating factor)
- Exact equations
- Bernoulli equations
- Substitution and homogeneous
- Initial value problems
Second-Order Linear
- Constant-coefficient homogeneous
- Characteristic equation, repeated roots
- Method of undetermined coefficients
- Variation of parameters
- Reduction of order
- Cauchy-Euler equations
Laplace Transforms
- Forward Laplace transform
- Inverse transform via tables
- Heaviside / step function
- Dirac delta function
- Convolution theorem
- Solving IVPs via Laplace
Systems & Phase Planes
- Linear systems with eigenvalues
- Phase plane analysis
- Equilibrium and stability
- Fundamental matrix
Series & Numerical
- Power series solutions
- Frobenius method
- Euler's method, improved Euler
- Runge-Kutta methods
- Slope fields
Applications
- Spring-mass systems (free, damped, forced)
- RLC circuits
- Population dynamics, logistic growth
- Mixing problems
- Newton's law of cooling
How I'd Walk You Through Choosing the Right First-Order Method
Problem: Solve dy/dx = (y² + 1) / x.
- Look for separability first. Always your first check. Can you write it as g(y) dy = f(x) dx? Here: dy / (y² + 1) = dx / x. Yes — separable.
- Integrate both sides. ∫ dy / (y² + 1) = ∫ dx / x gives arctan(y) = ln|x| + C.
- Solve for y if asked. y = tan(ln|x| + C).
- Why the flowchart matters. If separability had failed, the next checks are: linear (use integrating factor), exact (check ∂M/∂y = ∂N/∂x), Bernoulli (form y' + p(x)y = q(x)y^n), or substitution. We learn all five so you know which trap door to open.
The real lesson: First-order ODEs are 90% pattern recognition. Once you can classify the equation in 10 seconds, the actual solving is mechanical.
FAQ
What kinds of differential equations can you tutor?
All standard topics in a first ODE course: first-order (separable, linear, exact, Bernoulli), second-order linear (homogeneous and non-homogeneous), systems, Laplace transforms, series solutions, and applications.
Do you cover Laplace transforms?
Yes — Laplace transforms, inverse transforms, convolution, and applications to initial value problems are core to engineering ODE courses.
I'm in an engineering ODE course. Can you help with applications?
Yes. We connect every method to its application — RLC circuits, spring-mass systems, population dynamics — so the math has a purpose.