U1L19p1 Variation of Parameters | Method 4 | Second Order DE | 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 cover Method 4: Variation of Parameters – the most important method for solving Second Order Linear Differential Equations. Most AKTU exam questions come from this topic.
📌 General Form: d²y/dx² + P·dy/dx + Q·y = R
Variation of Parameters – Step by Step:
Step Description
1 Find C.F. = c₁·A + c₂·B (where A and B are functions of x)
2 Let P.I. = u·A + v·B
3 Compute u = -∫ (R·B) / (A·B' - A'·B) dx
4 Compute v = ∫ (R·A) / (A·B' - A'·B) dx
5 Complete solution: y = C.F. + P.I. = c₁A + c₂B + uA + vB
Note: If the question only says "Solve", you can use either C.F.+P.I. or Variation of Parameters. If the question explicitly says "by Variation of Parameters", you must use this method.
Solved Examples in this video:
Problem A B R Solution
d²y/dx² + a²y = sec(ax) cos(ax) sin(ax) sec(ax) y = c₁ cos ax + c₂ sin ax + (1/a²) cos ax·log(sec ax) + (x/a) sin ax
d²y/dx² - y = 2/(1+eˣ) eˣ e⁻ˣ 2/(1+eˣ) y = c₁eˣ + c₂e⁻ˣ + (complicated uA + vB)
0:00 – Method 4: Variation of Parameters (Most important topic)
0:13 – Basic concept and general form: d²y/dx² + P·dy/dx + Q·y = R
0:45 – Step 1: Find Complementary Function (C.F.)
1:34 – Step 2: P.I. = u·A + v·B
1:46 – Formulas for u and v:
3:00 – Example 1: d²y/dx² + a²y = sec(ax) (by Variation of Parameters)
3:42 – Step 1: Auxiliary Equation: m² + a² = 0 → m = ±ai
6:23 – Find u: u = -∫ (R·B) / a dx = -∫ (sec ax · sin ax) / a dx
7:05 – Find v: v = ∫ (R·A) / a dx = ∫ (sec ax · cos ax) / a dx
8:34 – Complete solution: y = C.F. + P.I.
9:11 – Example 2: d²y/dx² - y = 2/(1+eˣ) (by Variation of Parameters)
9:33 – Standard form: (D² - 1)y = 2/(1+eˣ)
10:36 – Find denominator: A·B' - A'·B
11:44 – Find u: u = -∫ (R·B) / (-2) dx = -∫ [2/(1+eˣ) · e⁻ˣ] / (-2) dx
13:49 – Find v: v = ∫ (R·A) / (-2) dx = ∫ [2/(1+eˣ) · eˣ] / (-2) dx
17:04 – Complete solution: y = C.F. + u·A + v·B
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?
• Complete explanation of Variation of Parameters method
• Clear formulas for u and v with integration
• Step-by-step solved examples (exam favorites)
• How to handle integration of rational/exponential functions
• Perfect for AKTU semester exam preparation
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