U1L3 nth Order Linear Differential Equations with Constant Coefficients | CF Tricks | BAS203 | AKTU

ENGINEERING MATHEMATICS-II (BAS203) | AKTU B.Tech 1st Year Sem 2
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In this video lecture with notes, we dive deep into nth Order Linear Differential Equations with Constant Coefficients – a very important topic for AKTU semester exam.
📌 Topics covered in this video:
• General form of nth Order Linear DE with Constant Coefficients
• Operator form (D-operator) and Auxiliary Form
• Auxiliary Equation (f(D) = 0)
• Complete Solution = Complementary Function (C.F.) + Particular Integral (P.I.)
• How to find Complementary Function (C.F.) for 3 cases:
1️⃣ Distinct Roots – C.F. = c₁e^(αx) + c₂e^(βx) + …
2️⃣ Equal Roots – C.F. = (c₁ + c₂x + c₃x² + …) e^(αx)
3️⃣ Imaginary (Complex) Roots (α ± iβ) – C.F. = e^(αx)[c₁ cos βx + c₂ sin βx]

This video is part of our ENGINEERING MATHEMATICS-II complete syllabus playlist for AKTU first year sem 2.
Useful for all branches – AKTU CSE, CS, IT, EC, EE, ME, CE – basically AKTU BTech first year students.
✅ Why watch?
• Simple step-by-step explanation
• Focus on AKTU important questions & exam pattern
• Easy tricks to remember CF for different root cases
• Perfect for AKTU exam preparation and AKTU study material
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0:00 – Introduction: nth Order Linear Differential Equations with Constant Coefficients
0:26 – General Form (a₀ dⁿy/dxⁿ + a₁ dⁿ⁻¹y/dxⁿ⁻¹ + … + aₙy = f(x))
1:00 – Meaning of nth Order, Linear, Constant Coefficients
2:07 – D-Operator Form (f(D) y = f(x))
3:48 – Auxiliary Form & Auxiliary Equation (f(D) = 0)
4:36 – Complete Solution = C.F. + P.I.
5:22 – How to find Complementary Function (C.F.)
5:53 – Case 1: Distinct Roots (C.F. = c₁e^(αx) + c₂e^(βx) + …)
6:41 – Example of Distinct Roots
7:20 – Case 2: Equal Roots (C.F. = (c₁ + c₂x + c₃x² + …) e^(αx))
8:08 – Case 3: Imaginary (Complex) Roots (α ± iβ)
8:46 – C.F. for Complex Roots: e^(αx)[c₁ cos βx + c₂ sin βx]
9:00 – Summary: 3 cases of roots

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