DIFFERENTIAL EQUATIONS & LAPLACE TRANSFORMS

1. Total Differential Equation (Pfaffian Differential Equations) 1-18

Introduction 1; Methods for Solving the Equation Pdx+Qdy+Rdz=0 1.  

2. Partial Differential Equations of the First Order, Lagrange's Equations, 

Charpit's General Method 19-89

Introduction 19; Partial Differential Equations 19; Order of Partial Differential Equations 19; Degree of the Partial Differential Equations 19; Linear Partial Differential Equations 20; Formation of a Partial Differential Equations 20; Formation of a Partial Differential Equation by Elimination of Arbitrary Constants 20; Formation of Partial Differential Equation by Elimination of Arbitrary Function f from the Equation f(u, v) = 0, where u, v are Functions of x, y, z 26; Solution of Partial Differential Equations 34; Lagrange's Method of Getting the General Solution in the Form f(u,v) = 0 35; General Solution of Lagrange's Equation 35; Some Special Types of Equations which can be Solved Easily by Methods other than the General Method 53; Standard Form I 53; Standard Form II 58; Standard Form III 64; Standard Form IV or Clairaut's Form 70; Charpit's Method 72; Compatible Differential Equations of First Order 85. 

UNIT-II

3. Linear Partial Differential Equations with Constant Coefficients 90-142

Introduction 90; Solution of Linear Partial Differential Equation 90; Complementary Solutions 90; When Auxiliary Equation has Two Equal Roots 92; Integration 99; Particular Integral (P.I.) 100; Short-cut Method 106; Particular Case, when F(a,b) = 0 112; General Method for Finding the Particular Integral 118; Non-homogeneous Linear Differential Equations 121; Particular Integrals (P.I.) 123; Partial Differential Equations Reducible to Equations with Constant Coefficients 136.

LAPLACE TRANSFORMS

UNIT-III

4. LAPLACE TRANSFORM 143-196 

Integral Transform 143; Laplace Transform 143; Properties of Laplace Transform 147; Laplace Transform of Discontinuous Functions 162; Existence Theorem of Laplace Transforms 166; Laplace Transform of Derivatives of F(t) 168; Differentiation of Laplace Transforms 169; Integration of Laplace Transforms 170; Initial Value Theorem 184; Final Value Theorem 185; Laplace Transform of Integrals 185; Evaluation of Integrals with the help of Laplace Transform 188; Periodic Function 194. 

5. THE INVERSE LAPLACE TRANSFORMS 197-250

Inverse Laplace Transform 197; Properties of Inverse Laplace Transform 198; Methods of Finding Inverse Laplace Transforms by Using Partial Fractions 214; Convolution 238; Convolution Theorem or Convolution Property 238.

UNIT-IV

6. APPLICATIONS OF LAPLACE TRANSFORMS 251-272

Solution of Linear Differential Equations with Constant Coefficients 251; Procedure for Application of Laplace Transform 251.

7. SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS 273-276

Definition 273; Theorem 273.