**Week 1** (Sections : (1.3, 2.1, 2.2, 2.4) **Basic mathematical models. ****Classification of differential equations. ****First order linear equations. Separable differential equations. Uniqueness of solution of linear first order ODE.**

**Basic mathematical models.**

**Classification of differential equations.**

**First order linear equations. Separable differential equations. Uniqueness of solution of linear first order ODE.****Week 2** (Sections : (2.4, 2.6, 2.8) Bernoulli equation. Exact differential equations and integrating factors. The existence and uniqueness theorem of the initial value problem for first order ODE’s.

**Week 3** (Sections : (2.8, 3.1, 3.2) The existence and uniqueness theorem (Picard’s iterations). Second order linear ODE’s with constant coefficients. Existence and uniqueness of solutions to the Cauchy problem. Abel’s theorem.

**Week 4** (Sections : (3.2, 3.3, 3.4)

## Applications of Abel’s theorem. Complex roots of characteristic equation. Reduction of order of second order linear ODE’s

**Week 5** (Sections : (3.6, 3.8, 5.1) Non-homogeneous second order ODE’s: method of variation of parameters. Forced periodic vibrations. Review of power series.

**Week 6** (Sections : (5.2, 5.3, 6.1) **Series solutions of ODEs. Laplace transform and its properties. **

**Week 7** (Sections : (6.2, 6.3, 6.4, 6.6. ) **Solution of initial value problems by using the Laplace transform. Step functions,. ODEs with discontinuous forcing functions. ****The convolution integral. **

**Week 8** (Sections : ( 7.1, 7.2, 7.3) **Systems of of first-order linear ODE’s. Matrices . Systems of linear algebraic equations. Eigenvalues and eigenvectors of matrices.**

**Week 9** (Sections : (7.4, 7.5, 7.6) Basic theory of systems of first-order linear ODE’s, Abel’s Theorem. Homogeneous linear systems with constant coefficients. Complex valued eigenvalues.

**Week 10** (Sections : (7.7, 7.8, 7.9) Fundamental Matrices. Repeated eigenvalues, Non-homogeneous systems.

**Week 11** (Sections : (10.1, 10.2, 10.3, 10.4) ,Two-point boundary value problems. Fourier series, The Fourier convergence Theorem. Even and odd functions (Fourier cosine and sine series).