Ordinary Differential Equations
In mathematics, a differential equation is an equation that relates one or more functions and their derivatives.
In the course on ordinary differential equations, we usually deal with the following topics:
Separable equations
First-order linear differential equations
Bernoulli differential equations
Exact differential equations and test for exactness
Homogeneous functions and equations
Riccati, Clairaut's, and Lagrange's equations
Orthogonal trajectories
The general form of a second-order linear differential equation
Second-order linear homogeneous differential equations with constant coefficients (real, complex, and repeated roots)
Reduction of order
Linear independence and Wronskian
Nonhomogeneous linear second-order differential equations
Undetermined coefficients and variation of parameters
Higher-order differential equations
Cauchy-Euler differential equations of order n
Power series solutions of differential equations
Regular singular points
The generalized power series method in solving differential equations
Definition of integral transforms
Definition of Laplace transform
A short table of Laplace and inverse Laplace transforms
Step functions and their Laplace transforms
Computation of Laplace transform of a function's derivative and antiderivative
Differentiation and integration of Laplace transforms
Dirac delta function
Convolution integral
Completion of the table of Laplace transforms
Systems of differential equations
Real, complex, and repeated eigenvalues and their applications in solving systems of differential equations
Nonhomogeneous systems of differential equations
For more details, refer to the following book:
Boyce, W. E., DiPrima, R. C., & Meade, D. B. (2017). Elementary differential equations. John Wiley & Sons.
For differential equations, a book (in Persian) by Dara Moazzami (pictures) has been also always very useful.