Contents

• Unit 1. Matrices: Rank of a matrix, rank-nullity theorem; System of linear equations; Symmetric, skew-symmetric and orthogonal matrices; Eigenvalues and eigenvectors; Diagonalization of matrices; Cayley-Hamilton Theorem, and Orthogonal transformation.

  Unit 2. First order ordinary differential equations: Linear and Bernoulli’s equations, Exact equations, Equations not of first degree: equations solvable for p,equations solvable for y, equations solvable for x and Clairaut’s type.

  Unit 3. Ordinary differential equations of higher orders: Linear Differential Equations of Higher order with constant coefficients, Simultaneous Linear Differential Equations, Second order linear differential equations with variable coefficients: Homogenous and Exact forms, one part of CF is known, Change of dependent and independent variables, method of variation of parameters, Cauchy- Euler equation; Power series solutions including Legendre differential equation and Bessel differential equations.

  Unit 4. Partial Differential Equations – First order: Order and Degree, Formation; Linear Partial differential equations of First order, Lagrange’s Form, Non Linear Partial Differential equations of first order, Charpit’s method, Standard forms.

  Unit 5. Partial Differential Equations– Higher order: Classification of Second order partial differential equations, Separation of variables method to simple problems in Cartesian coordinates including two dimensional Laplace, one dimensional Heat and one dimensional Wave equations.