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Differential Equations

Differential equations. Definitions. Order, degree. General, particular and singular solutions.

Differential equations of the first order and first degree. Methods of solution. Separation of variables. Homogeneous, exact and linear equations. Integrating factors. Bernoulli's equation.

Equations of the first order and higher degree, Clairaut's equation. Singular solutions and extraneous loci. Discriminant of a differential equation. p-discriminant, c-discriminant.

Second order differential equations

General solution of n-th order linear differential equations. Linearly dependent and independent sets of functions, Wronskian test for dependence.

Differential operators

Homogeneous linear differential equations with constant coefficients, Auxiliary equation, solutions

Methods for finding particular solutions of linear differential equations with constant coefficients. Method of Undetermined Coefficients, Variation of Parameters, Superposition. Operational methods.

Solutions to systems of simultaneous linear differential equations with constant coefficients

Solution of linear differential equations by power series. Solutions about ordinary points and singular points.

Special functions. Bessel's equation of order v. Bessel function of the first kind.