Applied Differential Calculus

1.- First order differential equations:
a. Linear equations.
b. Separable equations.
c. Exact equations.
d. Homogeneous equations.
e. Qualitative analysis of equations.
2.- Second order differential equations.
a. Nonlinear and linear equations.
b. Homogeneous and non-homogeneous Linear Equations.
c. Reduction of Order.
d. Euler-Cauchy Equations.
3.- Laplace transformations:
a. Definition.
b. Application to differential equations.
c. Convolution.
4.- Systems of differential equations:
a. Linear and Nonlinear Systems.
b. Vector representation.
c. Eigenvalues and linearization.
5. Fourier series and separation of variables:
a. Basic results.
b. Fourier Sine and Cosine Series.
c. Applications of Fourier series to differential equations.
6.- Numerical methods:
a. Euler method.
b. Runge-Kutta method.
c. Solution of boundary value problems.