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**On completion of this course, students are able to: **

• Use ordinary differential equations to model engineering phenomena such as electrical circuits, forced oscillation of mass-spring and elementary heat transfer.

• Use partial differential equations to model problems in fluid mechanics, electromagnetic theory, and heat transfer.

• Evaluate double and triple integrals to find the area, volume, mass and moment of inertia of plane and solid region.

• Use curl and divergence of a vector function in three dimensions, as well as apply the Green’s Theorem, Divergence Theorem and Stokes’ theorem in various applications like electricity, magnetism and fluid flow.

• Use Laplace transforms to determine general or complete solutions to linear ODE.

**The notes covers following concepts:**

** Unit-1:**

Differential equations-1:

Linear differential equations with constant coefficients: Solutions of second and higher order differential equations – inverse differential operator method, a method of undetermined coefficients and method of variation of parameters.

**Unit-2:**

Differential equations-2: Solutions of simultaneous differential equations of first order.

Linear differential equations with variable coefficients: Solution of Cauchy’s and Legendre’s linear differential equations.

Nonlinear differential equations – Equations solvable for p, equations solvable for y, equations solvable for x, general and singular solutions, Clairauit’s equations and equations reducible to Clairauit’s form.

**Unit-3:**

Partial Differential equations: Formulation of PDE by the elimination of arbitrary constants/functions, a solution of non-homogeneous PDE by direct integration, a solution of homogeneous PDE involving derivative with respect to one independent variable only. Derivation of one-dimensional heat and wave equations and their solutions by the variable separable method.

Double and triple integrals: Evaluation of double integrals. Evaluation by changing the order of integration and changing into polar coordinates. Evaluation of triple integrals.

**Unit-4:**

Integral Calculus: Application of double and triple integrals to find area and volume. Beta and Gamma functions, definitions, Relation between beta and gamma functions and simple problems.

Curvilinear coordinates Orthogonal curvilinear coordinates – Definition, unit vectors and scale factors. Expressions for gradient, divergence, and curl. Cylindrical and spherical coordinate systems.

**Unit-5:**

Laplace Transform:

Definition and Laplace transform of elementary functions. function – problems Inverse Laplace Transform: Inverse Laplace Transform – problems, Convolution theorem and problems, the solution of linear differential equations using Laplace Transforms.

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engineering mathematics 1st year-B.S.Grewal