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# CALCULUS AND DIFFERENTIAL EQUATIONS 21MAT11

VTU UNIVERSITY
Subject code
21MAT11
Semester
1st Sem

2021 Scheme M1 Notes – calculus and differential equations

## CALCULUS AND DIFFERENTIAL EQUATIONS21MAT11

### MODULE-1: DIFFERENTIAL CALCULUS - 1

Polar curves, the angle between the radius vector and the tangent, the angle between two curves. Pedal equations. Curvature and Radius of curvature – Cartesian, Parametric, Polar and Pedal forms. Problems.
Self-study: Center and circle of curvature, evolutes and involutes.

### MODULE-2: DIFFERENTIAL CALCULUS - 2

Taylor’s and Maclaurin’s series expansion for one variable (Statement only) – problems. Indeterminate forms-L’Hospital’s rule. Partial differentiation, total derivative-differentiation of composite functions. Jacobian and problems. Maxima and minima for a function of two variables Problems.
Self-study: Euler’s Theorem and problems. Method of Lagrange undetermined multipliers with a single constraint

### MODULE-3: ORDINARY DIFFERENTIAL EQUATIONS (ODE’S) OF FIRST ORDER

Linear and Bernoulli’s differential equations. Exact and reducible to exact differential equations. Applications of ODE’s-Orthogonal trajectories, Newton’s law of cooling. Nonlinear differential equations: Introduction to general and singular solutions; Solvable for p only; Clairaut’s equations, reducible to Clairaut’s equations. Problems.
Self-Study: Applications of ODE’s: L-R circuits. Solvable for x and y.

### MODULE-4: ORDINARY DIFFERENTIAL EQUATIONS OF HIGHER ORDER

Higher-order linear ODE’s with constant coefficients – Inverse differential operator, method of variation of parameters, Cauchy’s and Legendre homogeneous differential equations Problems.
Self-Study: Applications to oscillations of a spring and L-C-R circuits.

### MODULE-5: LINEAR ALGEBRA

Elementary row transformation of a matrix, Rank of a matrix. Consistency and Solution of a system of linear equations; Gauss-elimination method, Gauss-Jordan method and Approximate solution by Gauss-Seidel method. Eigenvalues and Eigenvectors-Rayleigh’s power method to find the dominant Eigenvalue and Eigenvector.
Self-Study: Solution of a system of equations by Gauss-Jacobi iterative method. The inverse of a square matrix by Cayley- Hamilton theorem.