U1L19p2 Variation of Parameters – Euler-Cauchy & Logarithm Problems | BAS203 | AKTU Sem 2

ENGINEERING MATHEMATICS-II (BAS203) | AKTU B.Tech 1st Year Sem 2

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In this video lecture with notes, we continue with Method 4: Variation of Parameters – now applying it to Euler-Cauchy (Homogeneous) equations and problems with logarithmic functions.
📌 Problems Covered (with hints):
Problem Type Key Steps
x²y" + 2xy' - 12y = x·log x Euler-Cauchy Convert using x = eᶻ, find C.F. = c₁x³ + c₂x⁻⁴, then apply Variation of Parameters
y" + 2y' + y = e⁻ˣ·log x Constant Coefficient C.F. = (c₁ + c₂x)e⁻ˣ, A = e⁻ˣ, B = x·e⁻ˣ, then use u, v formulas
y" - 3y' + 2y = sin(e⁻ˣ) Practice Problem Solve using Variation of Parameters
y" + 3y' + 2y = e^(eˣ) Practice Problem Solve using Variation of Parameters
0:00 – More Variation of Parameters problems
0:12 – Problem 1: x²y" + 2xy' - 12y = x·log x
5:32 – Problem 2: d²y/dx² + 2dy/dx + y = e⁻ˣ·log x
8:57 – Practice Problem 1: y" - 3y' + 2y = sin(e⁻ˣ)
9:14 – Practice Problem 2: y" + 3y' + 2y = e^(eˣ)
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?
• Variation of Parameters applied to Euler-Cauchy equations
• Step-by-step hints for complex logarithmic problems
• Important: how to handle A and B for repeated roots case
• Practice problems for self-study
• Perfect for AKTU semester exam preparation
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