1. Basic definitions - definition of the solution, general, particular, singular solutions, field of directions, Cauchy problem, boundary-value problem for differential equations. 2. Systems of nonlinear differential equations of the first order, conversion of one equation of the nth order to the system of the first order equations, Cauchy problem for systems of differential equations, existence and uniqueness of the solution, local and global properties. 3. Elementary methods for solving differential equations - equations with separable variables, homogeneous equations and some other types of the first order equations. 4. Linear equations of the first order - existence and uniqueness of the solution, solution to homogeneous and non-homogeneous equations, the formula for variation of constants. 5. Systems of linear equations of the first order - existence and uniqueness of the solution, fundamental solution matrix, the formula for variation of constants, systems of linear equations of the first order with constant coefficients. 6. Linear equations of the nth order, conversion to the systems of the first order linear equations, fundamental system of solutions, Wronskian, equations with constant coefficients, particular solution for special form of right-hand sides.
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Students will learn the basic topics of the theory of ordinary differential equations, the most important classes and methods for solving of differential equations. In the theoretical part, the existence and uniqueness theorem for non-linear equation and the conversion of the equation of the nth order to the system of the first order equations are included. Further, linear equations and systems of linear equations of the first order will be investigated. Methods of solving equations with separable variables, linear equations and systems of linear equations with constant coefficients, inclusive of variation of constants method, are involved.
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