DIFFERENTIAL EQUATIONS MTH401

Virtual University of Pakistan Knowledge beyond the boundaries

Table of Contents 1 Introduction ...... 1 2 Fundamentals ...... 3 2.1 Elements of the Theory ...... 3 2.2 Specific Examples of ODE’s ...... 3 2.3 The order of an equation ...... 4 2.4 Ordinary ...... 4 2.5 Partial Differential Equation ...... 4 2.6 Results from ODE data ...... 4 2.7 BVP Examples ...... 5 2.8 Properties of ODE’s ...... 5 2.9 Superposition ...... 5 2.10 Explicit Solution ...... 5 2.11 Implicit Solution ...... 5 3 Separable Equations ...... 6 3.1 Solution steps of Separable Equations ...... 6 3.2 Exercise ...... 13 4 Homogeneous Differential Equations ...... 15 4.1 Method of Solution...... 16 4.2 Equations reducible to homogenous form ...... 17 4.2.1 Case 1 ...... 18 4.2.2 Case 2 ...... 18 4.3 Exercise ...... 20 5 Exact Differential Equations ...... 22 5.1 Method of Solution ...... 22 5.2 Exercise ...... 26 6 Technique ...... 28 6.1 Case 1 ...... 28 6.2 Case 2 ...... 29 6.3 Case 3 ...... 29 6.4 Case 4 ...... 29 6.5 Exercise ...... 36 7 First Order Linear Equations ...... 37 7.1 Method of solution ...... 37 7.2 Exercise ...... 41 8 Bernoulli Equations ...... 43 8.1 Method of solution ...... 43 8.2 Exercise ...... 47 8.3 Substitutions ...... 48 8.4 Exercise ...... 52 9 Solved Problems ...... 53 10 Applications of First Order Differential Equations ...... 69 10.1 Orthogonal Trajectories ...... 69 10.2 Orthogonal curves ...... 71 10.3 Orthogonal Trajectories (OT) ...... 71 10.3.1 Method of finding Orthogonal Trajectory ...... 72 10.4 Population Dynamics ...... 77 11 Radioactive Decay ...... 80 11.1 Newton's Law of Cooling ...... 82 11.2 Carbon Dating ...... 84 12 Applications of Non-linear Equations ...... 86 12.1 Logistic equation ...... 86 12.1.1 Solution of the Logistic equation ...... 86 12.1.2 Special Cases of Logistic Equation...... 87 12.1.3 A Modification of LE...... 88 12.2 Chemical reactions ...... 89 12.3 Miscellaneous Applications ...... 92 13 Higher Order Linear Differential Equations ...... 94 13.1 Preliminary theory ...... 94 13.2 Initial -Value Problem ...... 94 13.2.1 Solution of IVP ...... 95 13.3 Theorem ( Existence and Uniqueness of Solutions) ...... 95 13.4 Boundary-value problem (BVP) ...... 97 13.4.1 Solution of BVP ...... 97 13.4.2 Possible Boundary Conditions ...... 98 13.5 Linear Dependence ...... 100 13.6 Linear Independence ...... 100 13.6.1 Case of two functions...... 100 13.7 ...... 102 13.8 Theorem (Criterion for Linearly Independent Functions) ...... 102 13.9 Exercise ...... 104 14 Solutions of Higher Order Linear Equations ...... 106 14.1 Preliminary Theory ...... 106 14.2 Superposition Principle...... 106 14.3 Linear Independence of Solutions ...... 109 14.4 Fundamental Set of Solutions ...... 109 14.4.1 Existence of a Fundamental Set ...... 110 14.5 General Solution-Homogeneous Equations...... 110 14.6 Non-Homogeneous Equations ...... 112 14.7 Complementary Function ...... 112 14.8 General Solution of Non-Homogeneous Equations ...... 113 14.9 Superposition Principle for Non-homogeneous Equations ...... 114 14.10 Exercise ...... 115 15 Construction of a Second Solution ...... 117 15.1 General Case ...... 117 15.2 Order Reduction ...... 120 15.3 Exercise ...... 123 16 Homogeneous Linear Equations with Constant Coefficients ...... 124 16.1 Method of Solution ...... 124 16.1.1 Case 1 (Distinct Real Roots) ...... 124 16.1.2 Case 2 (Repeated Roots) ...... 125 16.1.3 Case 3 (Complex Roots) ...... 125 16.2 Higher Order Equations ...... 127 16.2.1 Case 1 (Real distinct roots) ...... 127 16.2.2 Case 2 (Real & repeated roots) ...... 128 16.2.3 Case 3 (Complex roots) ...... 128 16.3 Solving the Auxiliary Equation ...... 128 17 Method of Undetermined Coefficients(Superposition Approach) ...... 132 17.1 The form of Input function g(x) ...... 132 17.2 Solution Steps ...... 133 17.2.1 Restriction on Input function g ...... 133 17.3 Trial particular solutions ...... 134 17.4 Input function g(x)as a sum ...... 134

17.5 Duplication between y p and yc ...... 138 17.6 Exercise ...... 144 18 Undetermined Coefficient (Annihilator Operator Approach)...... 145 18.1 Differential Operators ...... 145 18.2 Differential Equation in Terms of D ...... 146 18.3 Annihilator Operator ...... 148 18.4 Exercise ...... 153 19 Undetermined Coefficients(Annihilator Operator Approach) ...... 155 19.1 Solution Method ...... 155 19.2 Exercise ...... 165 20 ...... 166 20.1 First order equation ...... 167 20.2 Second Order Equation ...... 168 20.3 Summary of the Method ...... 170 20.3.1 Constants of Integration ...... 171 21 Variation of Parameters Method for Higher-Order Equations ...... 177 21.1 Exercise ...... 185 22 Applications of Second Order Differential Equation...... 186 22.1 Simple Harmonic Motion ...... 186 22.1.1 Hook’s Law ...... 186 22.1.2 Newton’s Second Law ...... 187 22.1.3 Weight ...... 187 22.1.4 Differential Equation ...... 187 22.1.5 Initial Conditions ...... 188 22.1.6 Solution and Equation of Motion ...... 188 22.1.7 Alternative form of Solution ...... 189 22.1.8 Amplitude ...... 189 22.1.9 A Vibration or a Cycle ...... 189 22.1.10 Period of Vibration ...... 190 22.1.11 Frequency ...... 190 22.2 Exercise ...... 196 23 Damped Motion ...... 197 23.1 Damping Force ...... 197 23.2 The Differential Equation ...... 197 23.2.1 Solution of the Differential Equation ...... 198 23.2.2 Alternative form of the Solution ...... 200 23.2.3 Quasi Period ...... 209 23.3 Exercise ...... 209 24 Forced Motion ...... 211 24.1 Forced motion with damping ...... 211 24.2 Transient and Steady-State Terms ...... 214 24.3 Motion without Damping ...... 217 24.4 Electric Circuits ...... 219 24.5 The LRC Series Circuits ...... 219 24.5.1 Resistor ...... 219 24.5.2 Inductor ...... 220 24.5.3 Capacitor ...... 220 24.6 Kirchhoff’s Voltage Law ...... 220 24.6.1 The Differential Equation ...... 221 24.6.2 Solution of the differential equation ...... 221 Case 1 Real and distinct roots ...... 222 Case 2 Real and equal ...... 222 Case 3 Complex roots ...... 222 25 Forced Motion (Examples) ...... 224 26 Differential Equations with Variable Coefficients ...... 230 26.1 Cauchy- Euler Equation ...... 230 26.1.1 Method of Solution ...... 231 26.1.2 Case-I (Distinct Real Roots) ...... 231 26.1.3 Case II (Repeated Real Roots) ...... 232 26.1.4 Case III (Conjugate Complex Roots) ...... 233 26.2 Exercises ...... 235 27 Cauchy-Euler Equation (Alternative Method of Solution) ...... 237 27.1 Exercises ...... 244 28 Power Series (An Introduction) ...... 245 28.1 Power Series ...... 245 28.2 Convergence and Divergence ...... 245 28.2.1 The Ratio Test ...... 246 28.2.2 Interval of Convergence ...... 246 28.2.3 Radius of Convergence ...... 246 28.2.4 Convergence at an Endpoint ...... 247 28.3 Absolute Convergence ...... 248 28.4 Power Series Representation of Functions ...... 249 28.4.1 Theorem ...... 249 28.4.2 Series that are Identically Zero ...... 250 28.5 Analytic at a Point ...... 250 28.6 Arithmetic of Power Series ...... 251 29 Power Series Solution of a Differential Equation ...... 254 29.1 Exercise ...... 258 30 Solution about Ordinary Points ...... 259 30.1 Analytic Function ...... 259 30.2 Ordinary and singular points ...... 259 30.2.1 Polynomial Coefficients...... 259 30.3 Theorem (Existence of Power Series Solution) ...... 260 30.4 Non-polynomial Coefficients ...... 264 30.5 Exercise ...... 265 31 Solutions about Singular Points ...... 266 31.1 Regular and Irregular Singular Points ...... 266 31.1.1 Polynomial Coefficients...... 266 31.2 Method of Frobenius ...... 268 31.2.1 Frobenius’ Theorem ...... 268 31.3 Cases of Indicial Roots ...... 273 31.3.1 Case I (Roots not Differing by an Integer) ...... 273 32 Solutions about Singular Points ...... 275 32.1 Method of Frobenius (Cases II and III) ...... 275 32.1.1 Case II (Roots Differing by a Positive Integer) ...... 275 33 Bessel’s Differential Equation ...... 285 33.1 Series Solution of Bessel’s Differential Equation ...... 285 33.2 Bessel’s Function of the First Kind ...... 287 34 Legendre’s Differential Equation ...... 294 34.1 Legendre’s Polynomials ...... 296 34.2 Rodrigues Formula for Legendre’s Polynomials...... 297 34.3 Generating Function For Legendre’s Polynomials ...... 297 34.4 ...... 298 34.5 Orthogonally of Legendre’s Polynomials...... 299 34.6 Normality condition for Legendre’ Polynomials ...... 301 34.7 Exercise ...... 303 35 Systems of Linear Differential Equations ...... 304 35.1 Simultaneous Differential Equations ...... 304 35.2 Solution of a System ...... 305 35.2.1 Systematic Elimination (Operator Method) ...... 305 36 Systems of Linear Differential Equations ...... 312 36.1 Solution of Using Determinants ...... 312 36.2 Solution Method ...... 312 36.3 Exercise ...... 320 37 Systems of Linear First-Order Equation ...... 321 37.1 The nth Order System ...... 321 37.2 Linear Normal Form ...... 321 37.3 Reduction of a Linear Differential Equation to a System ...... 322 37.3.1 Systems Reduced to Normal Form ...... 325 37.4 Degenerate Systems ...... 328 37.5 Applications of Linear Normal Forms...... 329 38 Introduction to Matrices ...... 332 38.1 Matrix ...... 332 38.2 Rows and Columns ...... 332 38.3 Order of a Matrix ...... 332 38.4 Square Matrix ...... 332 38.5 Equality of matrix ...... 332 38.6 Column Matrix ...... 332 38.7 Multiple of matrices...... 333 38.8 Addition of Matrices ...... 334 38.9 Difference of Matrices ...... 334 38.10 Multiplication of Matrices ...... 335 38.11 Multiplicative Identity ...... 337 38.12 Zero Matrix ...... 337 38.13 Associative Law ...... 338 38.14 Distributive Law ...... 338 38.15 Determinant of a Matrix ...... 338 38.16 Transpose of a Matrix ...... 338 38.17 Multiplicative Inverse of a Matrix ...... 340 38.18 Non-Singular Matrices ...... 340 38.19 Derivative of a Matrix of functions ...... 343 38.20 of a Matrix of Functions ...... 343 38.21 Augmented Matrix ...... 344 38.22 Elementary Row Operations ...... 345 38.23 The Gaussian and Gauss-Jordon Methods ...... 345 38.24 Exercise ...... 349 39 The Eigenvalue problem ...... 351 39.1 Eigenvalues and Eigenvectors ...... 351 39.2 The Non-trivial solution ...... 351 39.3 Exercise ...... 357 40 Matrices and Systems of Linear First-Order Equations ...... 358 40.1 Matrix form of a system ...... 358 40.2 Initial –Value Problem...... 361 40.3 Theorem: Existence of a unique Solution...... 362 40.4 Superposition Principle...... 362 40.5 Linear Dependence of Solution Vectors ...... 364 40.6 Linear Independence of Solution Vectors ...... 365 40.7 Exercise ...... 366 41 Matrices and Systems of Linear 1st-Order Equations (Continued) ...... 369 41.1 Theorem ...... 369 41.2 Fundamental set of solution ...... 370 41.2.1 Theorem (Existence of a Fundamental Set) ...... 370 41.3 General solution ...... 370 41.4 Non-homogeneous Systems ...... 372 41.4.1 Particular Integral...... 372 41.5 Theorem ...... 373 41.5.1 Complementary function ...... 373 41.5.2 General solution of a Non homogenous systems ...... 373 41.6 Fundamental Matrix ...... 374 41.7 Exercise ...... 375 42 Homogeneous Linear Systems ...... 377 42.1 Eigenvalues and Eigenvectors ...... 379 42.1.1 Case 1 (Distinct real eigenvalues)...... 380 42.1.2 Case 2 (Complex eigenvalues)...... 385 42.2 Theorem (Solutions corresponding to complex eigenvalues ) ...... 386 42.3 Theorem(Real solutions corresponding to a complex eigenvalue) ...... 388 42.4 Exercise ...... 390 43 Real and Repeated Eigenvalues ...... 392 43.1 Eigenvalue of multiplicity m ...... 392 43.1.1 Method of solution ...... 392 43.1.2 Eigenvalue of Multiplicity Two ...... 393 43.1.3 Eigenvalues of Multiplicity Three ...... 399 44 Non-Homogeneous System ...... 403 44.1 Definition ...... 403 44.2 Matrix Notation ...... 403 44.3 Method of Solution ...... 403 44.4 Method of Undetermined Coefficien