U1L18p1 Changing of Independent Variable | Second Order DE | Method 3 | 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 3: Changing of Independent Variable for solving Second Order Linear Differential Equations – a very important topic for AKTU semester exams (most questions come from Methods 3 & 4).
📌 General Form: d²y/dx² + P·dy/dx + Q·y = R
After changing independent variable x → z, the equation becomes:
d²y/dz² + P₁·dy/dz + Q₁·y = R₁
Formulas for P₁, Q₁, R₁:
Formula Expression
P₁ (d²z/dx² + P·dz/dx) / (dz/dx)²
Q₁ Q / (dz/dx)²
R₁ R / (dz/dx)²
Solved Example in this video:
• x d²y/dx² - dy/dx - 4x³y = 8x³ sin(x²)
0:00 – Method 3: Changing of Independent Variable
0:26 – General form: d²y/dx² + P·dy/dx + Q·y = R
1:44 – Formulas for P₁, Q₁, R₁:
1:57 – P₁ = (d²z/dx² + P·dz/dx) / (dz/dx)²
2:06 – Q₁ = Q / (dz/dx)²
2:17 – R₁ = R / (dz/dx)²
2:35 – Strategy: First find Q₁ = constant (choose wisely)
2:51 – Solved Example: x d²y/dx² - dy/dx - 4x³y = 8x³ sin(x²)
4:43 – Step 1: Find Q₁ = constant
8:38 – Step 2: Find P₁
9:48 – Step 3: Find R₁
10:46 – Step 4: Write transformed equation
13:45 – Final Answer: y = c₁e^(x²) + c₂e^(-x²) - sin(x²)
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 Changing of Independent Variable method
• Clear formulas for P₁, Q₁, R₁
• Step-by-step solved example (exam favorite)
• How to choose the constant for Q₁
• Reduction to constant coefficient DE
• Perfect for AKTU semester exam preparation
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