Unit 2 & 3, Differential Equations 2 & 3 Notes by
Unit 1, Differential Equations – I and Unit 4, Partial Differential Equations (PDE), Notes by
Unit 7, Laplace Transforms – I and Unit 8, Laplace Transforms – II by
Unit 5, Integral Calculus, Unit 5A, Beta and Gamma Functions and Unit 6, Vector Integration by
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Differential Equation- I
Equations of first order and higher degree
In this unit, we shall study differential equations of the first order and higher degree, We study the differential equations solvable for and the problems involving in it, differential equation solvable for and the problems involving in it, Differential equation solvable for and some problems involving in it. We discuss the problems on special type called Clairaut’s Equation and reducible to clairaut’s form involving both general solution and singular solution and we discuss the application of the first order and first degree differential equation with illustrative examples.
At the end of this unit he will be able to understand
• To obtain the solution of non-linear differential equation.
• To obtain the solution of the differential equation of the form .
• The method of the solution is simple involving well known methods.
• Singular solution exists for higher degree equations of first order.
• Clairaut’s equation has numerous engineering applications like geodesics.
• Non linear equation of first order differential equation is reduced to linear differential equations of first order.
• Mathematical models for some of the applications like Kirchoff’s law, Newton’s law of cooling etc.
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PARTIAL DIFFERENTIAL EQUATIONS
In this unit we study how to form a P.D.E and various methods of obtaining solutions of P.D.E. This unit consists of 6 sections.
In section 1, we learn how to form the P.D.E. by eliminating arbitrary constants and in section 2 we learn the formation of P.D.E by eliminating arbitrary functions. In section 3, the solution of non homogeneous P.D.E by the method of direct integration is discussed. In section 4, the solution of homogeneous equations is discussed. In section 5 we learn the method of separation of variables to solve homogeneous equations. In section 6 we discuss the Lagrange’s linear equation and the solution by the method of grouping and multipliers, at end some muliple choice questions prominence the comprehensive unit.
At the finish of this unit, we will be able to:
• Form Partial differential equation.
• Solve the first order linear partial differential equation
• Obtain the solution of homogeneous P.D.E by different methods.
• Obtain the solution of non homogeneous P.D.E
Formation of P.D.E by eliminating arbitrary constants
At the closing stages of this Section, we will be able to recognize:
• To identify P.D.E order, degree and classification of a P.D.E.
• Formation of P.D.E by elimination of arbitrary constants.
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