U1L16p1 Second Order Linear DE | Changing of Dependent Variable | Method 1 | 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 begin a new topic: Second Order Linear Differential Equations – an important topic for AKTU semester exams.
📌 General Form of Second Order Linear DE:
d²y/dx² + P·dy/dx + Q·y = R
where P, Q, R are functions of x only.

Method 1: Changing of Dependent Variable
• Let the solution be y = u·v
• u = Complementary Function (C.F.) found using conditions below
• v = function to be determined by substitution
Conditions for finding u (C.F.):
Solved Example in this video:
• d²y/dx² - cot x dy/dx - (1 - cot x)y = eˣ·sin x
• Step-by-step: Find P, Q, R → Check conditions → u = eˣ → Let y = eˣ·v → Substitute →

Reduce to first order linear DE → Solve

0:00 – Introduction: Second Order Linear Differential Equations
1:22 – Four Methods to solve Second Order DEs:
1:31 – Method 1: Changing of Dependent Variable
1:49 – Method 2: Normal Form
2:03 – Method 3: Changing of Independent Variable
2:20 – Method 4: Method of Variation of Parameters
3:33 – Detailed: Changing of Dependent Variable
5:46 – Conditions to find u (C.F.):
6:07 – Condition 1: 1 + P + Q = 0 → u = eˣ
6:38 – Condition 2: 1 - P + Q = 0 → u = e⁻ˣ
6:57 – Condition 3: P + Qx = 0 → u = x
7:22 – Condition 4: 2 + 2Px + Qx² = 0 → u = x²
7:48 – Condition 5: 1 + P/a + Q/a² = 0 → u = eᵃˣ
13:01 – Solved Example:
13:09 – d²y/dx² - cot x dy/dx - (1 - cot x)y = eˣ·sin x

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?
• Introduction to Second Order Linear DEs
• Complete explanation of Changing of Dependent Variable method
• Conditions to find u (C.F.) – exam-focused
• Step-by-step solution of a complex example
• Reduction from second order to first order
• Perfect for AKTU semester exam preparation
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