Higher Order Differential Equations – Roots & C.F. | (D⁴ + 64)y = 0

Solution of (D⁴ + 64) y = 0 (For BMATE301 / BEE301 – Module 1, BMATEC301 / BEC301 – Module 4) (2025 Scheme – 1BMATE101 / 1BMATC101 – Module 4) (2022 Scheme – BMATM101 / BMATC101 – Module 4) Step 1 – Auxiliary equation: m⁴ + 64 = 0 Step 2 – Factorize using sum of squares: m⁴ + 64 = (m² + 8)(m² − 8) = (m² + 8)(m − 2√2)(m + 2√2) Step 3 – Roots: m = ±i√8 = ±2√2 i (complex) m = ±2√2 (real) Step 4 – Complementary function (C.F.): For real roots ±2√2 → e^(2√2 x), e^(−2√2 x) For complex roots ±2√2 i → cos(2√2 x), sin(2√2 x) Step 5 – General solution: y = C₁ e^(2√2 x) + C₂ e^(−2√2 x) + C₃ cos(2√2 x) + C₄ sin(2√2 x) #HigherOrderDifferentialEquations #VTU #BMATE301 #BEE301 #BMATEC301 #BEC301 #EngineeringMathematics #VTUMaths #DifferentialEquations #CFandPI #VTUExam #MathsForEngineers #VTUModule1 #VTUModule4 #VTUSolutions #EngineeringMaths3 #VTUStudyMaterials #MathsLecture #AuxiliaryEquation #ComplementaryFunction #VTUMathsSolution #1BMATE101 #1BMATC101 #BMATM101 #BMATC101