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#### Apply Ordinary Differential Equations to Model and Solve Real-World Problems in Physics, Engineering, and Biology

The Ordinary Differential Equations (ODEs) course introduces students to the theory and techniques of solving and analyzing ordinary differential equations. It covers first-order and second-order ODEs, exploring separable, linear, and exact equations, as well as series solutions. Students learn Laplace transforms and their applications to solve ODEs and understand stability analysis for systems of ODEs. Real-world applications are emphasized throughout, including problems in science and engineering, such as heat conduction and population dynamics. Numerical methods may be introduced to approximate solutions when analytical methods are challenging.

By the end of the course, students gain proficiency in solving ODEs and using them to model various natural phenomena, enabling them to apply these skills to a wide range of practical problems in various disciplines.

#### Course Highlights

• Introduction to Differential Equations:
• First-Order and Second-Order Linear ODEs:
• Series Solutions of ODEs:
• Laplace Transform and ODEs:
• Systems and Applications of ODEs:
• Numerical Methods for ODEs (possible inclusion, depending on the course):

#### Course Benefits

• Analyze and classify different types of ordinary differential equations based on their order and linearity.
• Solve first-order ODEs using various techniques including separation of variables, integrating factors, and exact equations.
• Solve second-order linear homogeneous and non-homogeneous ODEs with constant coefficients using undetermined coefficients and variation of parameters.
• Apply power series and Frobenius methods to solve ODEs with non-constant coefficients.
• Use Laplace transforms to solve initial value problems for linear constant coefficient ODEs.
• Analyze and solve systems of first-order ODEs using matrix methods and understand stability and equilibrium points.
• Apply ODEs to model and solve real-world problems such as spring-mass systems, electrical circuits, population dynamics, and heat conduction.
• Utilize numerical methods to approximate solutions to ODEs when analytical techniques are challenging or impractical.
• Demonstrate proficiency in mathematical reasoning and problem-solving skills in the context of ordinary differential equations.

Course Typically Offered: Online every quarter.

Software Requirement: Students will use a graphing calculator. Many options are available online for free. An example is Desmos: https://www.desmos.com/calculator

Prerequisites:  Before taking this course, students should have previously taken Calculus I and Calculus II, or an equivalent course.

More Information: For more information about this course contact unex-techdata@ucsd.edu.

Course Number: MATH-40026
Credit: 4.00 unit(s)

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