# Elementary Differential Equations with Applications – Course Outlines

## Objective:

The word differential and equations certainly suggest solving some kind of equation that contains derivatives. Why should an erstwhile scientist or engineer study this subject? The answer is simple: Differential equations are the mathematical backbone of many areas of science and engineering in which we examine, albeit briefly, how differential equations arise from attempts to formulate or describe, certain physical system in term of mathematics.

## Recommended Books for Elementary Differential Equations with Applications

• Dennis G. Zill,  “Differential Equations with boundary- value problems”  3rd Edition.
• William E.Boyce and Richard C.Diprima, “ Elementary Differential Equations and boundary- value problems”  5th  Edition, published by John Wiley and Sons,1992.
• Differential Equations with Applications and Historical Notes By George F.Simmons,2nd Edition published by McGraw-Hill,1991.

## Course Outline of Elementary Differential Equations with Applications

Definitions, Existence, and uniqueness of the solution. First-order and simple higher-order ordinary differential equations (ode). Linear and nonlinear 1st order ode. Clauriat’s Equation. Ricatti equation Higher-order differential equations. UC- Methods and method of variation of parameters. Cauchy Euler’s equations. Laplace transforms

List of Topics to be covered

Week 1:

Introduction,  differential equations and their classification, Basic definitions. first-order differential equations, Separable differential equations.

Week 2:

Formation of a differential equation, Solution of IVP. Separable and nonseparable differential equations. Introduction to homogeneous functions. Solutions of homogeneous differential equations by substitution.

Week 3:

Solutions of nonhomogeneous differential equations reducible to homogeneous differential equations. Exact equation of nonhomogeneous differential equations, Non-exact ordinary differential equations.

Week 4:

The solution of ODE by the integration factor for homogeneous differential equations.

The solution of ODE by the integration factor for non-homogeneous differential equations.

Week 5:

Linear differential equations. Non-linear differential equations (Bernoulli Equation).

Week 6 :

Clauriat’s Equation. Ricatti equation.

Week 7:

The trajectory of orthogonal curves(in cartesian & polar coordinates).

Week 8:

Applications of first-order differential equations.

Week 9:

Second-order homogeneous differential equations with constant coefficients with distinct and repeated roots, Complimentary, Particular and General solution of homogeneous second order differential equations with complex roots.

Week 10:

Introduction of forcing functions General solution of non-homogeneous second order differential equations with distinct and repeated & complex roots.

Week 11:

Methods of undetermined coefficients (UC-method).

Week 12:

Variation of parameters.Cauchy-Euler equations.

Week 13:

Mechanical and electrical vibrations and other applications to mathematical modeling.Total differential Equations.

Week 14:

Laplace Transforms & its properties.

Week 15:

Application to IVP.

Week 16:

Power series solutions.

Prof.Fazal Rehman Shamil (Available for Professional Discussions)
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