Higher order differential equations engineering mathematics 🚀Finding Complementary factor C.F VTU

In today's video we will learn to find the complementary factor of higher order differential equations in Engineering Mathematics for the VTU syllabus. Welcome to the ultimate guide on finding the Complementary Factor (CF) for Higher Order Differential Equations! 🚀 Problems to Solve – Finding CF (Complementary Function) 🧠We'll solve a variety of problems – more than enough to make you perfect in finding the CF for any scenario! 1)d²y/dx² + 3(dy/dx) + 2y = 0 2)d²y/dx² - 4y = 0 3)(D² + 9)y = 0 4)y'' - 4y' + 13y = 0 5)(D³ + 8)y = 0 6)(D⁴ + 64)y = 0 As you know, the complete solution to a higher-order linear differential equation is: y=Complementary Factor (CF)+Particular Integral (PI) The CF depends entirely on the Left-Hand Side (LHS) of the equation (the homogeneous part). This video is dedicated to mastering the CF! If the Right-Hand Side (RHS) of the D.E. is zero, then the complete solution is simply y=CF. 🎯To find the Complementary Factor (CF), we must first find the roots of the Auxiliary Equation. The form of the CF depends entirely on the nature of these roots. Case 1: Roots are Real and Distinct (Non-repeated) Rule: If the roots are m1, m2, m3, ... which are all real and different, the Complementary Factor (CF) is: CF = C1 e^(m1 x) + C2 e^(m2 x) + C3 e^(m3 x) + ... Example: If roots are 1, 2, and 3. CF = C1 e^(1x) + C2 e^(2x) + C3 e^(3x) Case 2: Roots are Real and Repeated Rule: If a real root, say 'm', is repeated k times, the contribution to the CF is: CF_repeated = (C1 + C2 x + C3 x^2 + ... + Ck x^(k-1)) e^(m x) Example A (Repeated Twice): If roots are 2 and 2 (repeated twice, k=2). CF = (C1 + C2 x) e^(2x) Example B (Repeated Three Times): If roots are 3, 3, 3 (repeated three times, k=3). CF = (C1 + C2 x + C3 x^2) e^(3x) Case 3: Roots are Complex Conjugate Rule: If the roots are a pair of complex conjugates, m1 = alpha + ibeta and m2 = alpha - ibeta (where alpha is the real part and beta is the imaginary part), the Complementary Factor (CF) is: CF = e^(alpha x) [C1 cos(beta x) + C2 sin(beta x)] Example: If roots are 1 + 4i and 1 - 4i. Here, alpha = 1 and beta = 4. CF = e^(1x) [C1 cos(4x) + C2 sin(4x)] higher order differential equations engineering mathematics higher order differential equations engineering mathematics vtu complementary function of differential equations Please go through the playlists given below for Chapter wise videos on Engineering Mathematics ,Polytechnic diploma and other subjects for all universities across India and abroad. Engineering mathematics 1 playlist    • 21MAT21 VTU ADVANCED CALCULUS AND NUMERICA...   Engineering mathematics 2 playlist    • 21MAT11 | CALCULUS AND DIFFERENTIAL EQUATI...   ...................................................................................................................... 📍 Useful Links: My website https://mudassiracademy.com/ https://engineeringtuitionbangalore.com/ Facebook   / themudassiracademy   Twitter   / muddasiracademy   Instagram   / themudassiracademy