Pathway 03 · University
Differential Equations
Ordinary differential equations from first-order equations through linear constant-coefficient equations, systems and Laplace transform methods, ending with initial and boundary value problems. Every method is paired with a way to check its answer.
- Audience
- University students in a first ODE course, and engineering or physics students who need to solve and interpret differential equations in their own subjects.
- Modules
- 6, in order
- Level
- University
Prerequisites
- Single-variable integration techniques
- Complex numbers in the form and Euler's formula
- Matrices, determinants and solving linear systems
You will be able to
- Solve separable and first-order linear equations and initial value problems, and verify each solution by substitution
- Solve linear constant-coefficient equations of any order from the characteristic polynomial, including repeated and complex roots
- Find particular solutions by undetermined coefficients and variation of parameters, including resonant forcing
- Solve linear systems using eigenvalues and eigenvectors, and sketch the phase portrait of a system
- Solve initial value problems with discontinuous or impulsive forcing using the Laplace transform
- Solve simple two-point boundary value problems and identify cases with no solution or infinitely many
Modules
Work through these in order.
- 01
First-order equations
Classify first-order equations and solve the main solvable types.
- 02
Linear equations with constant coefficients
Solve homogeneous linear equations of any order through the characteristic polynomial.
- 03
Nonhomogeneous equations
Construct particular solutions and understand when the standard trial functions fail.
- 04
Systems of linear equations
Write higher-order equations as first-order systems and solve them with linear algebra.
- 05
Laplace transform methods
Turn initial value problems into algebra in the transform variable and invert the result.
- 06
Initial and boundary value problems
Contrast initial value problems with two-point boundary value problems and their different behavior.