Hindi - Himachal Pradesh

Second order partial differential equations: Origin and classification of second orderpartial differential equation, solution of linear partial differential equation with constantcoefficients. Monge’s method to solve the non-linear partial differential equationRr+Ss+Tt = V.Laplace Transforms:Introduction, basic theory of Laplace transforms, solution of initial value problem usingLaplace transforms, shifting theorems, unit step function, Dirac-delta function.Differentiation and integration of Laplace transforms. Convolution theorem.Calculus of Variations:Variation problems with fixed boundaries-Euler’s equation for functionals containing firstorder derivative and one independent variable. Extremals. Functionals dependent onhigher order derivatives. Functionals dependent on more than one independent variable.Variational problems in parametric form. Invariance of Euler’s equation under coordinatestransformation.Variational problems with moving boundaries-functionals dependent on one and twofunctions.Sufficient conditions for an Extremum-Jacobi and Legendre conditions.Numerical Analysis and computer programming:Numerical Methods: Solution of algebraic and transcendental equations of one variable byBisection, Secant, Regula Falsi, Newton-Raphson Method, Roots of Polynomials.Linear Equations: Solution of system of linear equations by Gaussian elimination method,Gauss-Seidel iterative method.Interpolation: Lagrange and Newton interpolation, divided differences, differenceschemes, interpolation formulas using differences.Numerical Differentiation: Solution of ordinary differential equations by Euler’s method,Runge-Kutta’s II and IV order method.Numerical Integration: Simpson’s 1/3 rule, Simpson’s 3/8 rule, Trapezodial rule, Gaussianquadrature formula.Programming in C: Algorithms and flow-charts for solving numerical problems.Developing simple programs in C language for problems involving techniques covered inthe numerical analysis.MECHANICAL ENGINEERINGPAPER-IEngineering Mechanics: Free body diagrams and equilibrium; centre of gravity andmoment of inertia, trusses and frames; principal of virtual work; kinematics and dynamics ofparticles and of rigid bodies in plane motion, including impulse and momentum (linear andangular) and energy formulations; impact.Strength of Materials: Stress and strain, stress-strain relationship and elastic constants,Mohr’s circle for plane stress and plane strain, thin cylinders; shear force and bending momentdiagrams; bending and shear stresses; combined bending and direct stresses, slope anddeflection of cantilever and simply supported beams with point loads and UDL; torsion ofcircular shafts; Euler’s theory of columns; strain energy methods; thermal stresses, failuretheories, unsymmetrical bending and shear centre, theories of elastic failure.