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Cauchy Euler DE.pptx
- 1. Cauchy – EulerHomogeneous Differential Equations 21UMT2T431: Ordinary Differential Equations And Real Analysis – I Dr. Soya Mathew Assistant Professor in Mathematics Department of Physical Sciences Kristu Jayanti College (Autonomous) Bengaluru – 560077, India
- 2. Cauchy – EulerDifferential Equations A Linear Differential Equation of the form 𝒙𝒏 𝒅𝒏𝒚 𝒅𝒙𝒏 + 𝒂𝟏𝒙𝒏−𝟏 𝒅𝒏−𝟏𝒚 𝒅𝒙𝒏−𝟏 + ⋯ + 𝒂𝒏𝒚 = 𝒇 𝒙 where 𝑎1, 𝑎2, … 𝑎𝑛 are constants and 𝑓 𝑥 is a known function of the 𝑥 is called a called Cauchy – Euler Homogeneous Differential Equation of order 𝑛.
- 3. Working Rule Given 𝑥𝑛 𝐷𝑛 +𝑎1𝑥𝑛−1 𝐷𝑛−1 + ⋯ + 𝑎𝑛 𝑦 = 𝑓 𝑥 where 𝐷 = 𝑑 𝑑𝑥 Put 𝑥 = 𝑒𝑧 ⟹ 𝑧 = log 𝑥 Then 𝑥𝐷 = 𝐷1 where 𝐷1 = 𝑑 𝑑𝑧 𝑥2𝐷2 = 𝐷1(𝐷1 − 1) 𝑥3 𝐷3 = 𝐷1(𝐷1 − 1)(𝐷1 − 2) and so on. Hence the equation becomes, 𝐹 𝐷1 𝑦 = 𝑓1 𝑧 where 𝑓1(𝑧) is the value of 𝑓(𝑥) in terms of 𝑧.
- 4. Problem: Solve 𝟒𝒙𝟐𝒅𝟐𝒚 𝒅𝒙𝟐 + 𝟒𝒙 𝒅𝒚 𝒅𝒙 − 𝒚 = 𝟒𝒙𝟐 Solution: Given 4𝑥2 𝑦′′ + 4𝑥𝑦′ − 𝑦 = 4𝑥2 ⟹ (4𝑥2 𝐷2 + 4𝑥𝐷 − 1)𝑦 = 4𝑥2 … … 1 where 𝐷 = 𝑑 𝑑𝑥 This is a second order Cauchy – Euler equation. Put 𝑥 = 𝑒𝑧 ⟹ 𝑧 = log 𝑥 Then 𝑥𝐷 = 𝐷1, 𝑥2 𝐷2 = 𝐷1 𝐷1 − 1 , where 𝐷1 = 𝑑 𝑑𝑧
- 5. Problem: Solve 𝟒𝒙𝟐𝒅𝟐𝒚 𝒅𝒙𝟐 + 𝟒𝒙 𝒅𝒚 𝒅𝒙 − 𝒚 = 𝟒𝒙𝟐 (Cont…) 1 ⟹ 4𝐷1 𝐷1 − 1 + 4𝐷1 − 1 𝑦 = 4 𝑒2𝑧 ⟹ 4𝐷1 2 − 4𝐷1 + 4𝐷1 − 1 𝑦 = 4 𝑒2𝑧 ⟹ 4𝐷1 2 − 1 𝑦 = 4 𝑒2𝑧 Auxiliary Equation (AE) is 4𝑚2 − 1 = 0 ⟹ 2𝑚 + 1 2𝑚 − 1 = 0
- 6. Problem: Solve 𝟒𝒙𝟐𝒅𝟐𝒚 𝒅𝒙𝟐 + 𝟒𝒙 𝒅𝒚 𝒅𝒙 − 𝒚 = 𝟒𝒙𝟐 (Cont…) ⟹ 𝑚 = − 1 2 , 1 2 The complementary function is 𝑐1𝑒− 𝑧 2 + 𝑐2𝑒 𝑧 2. ⟹ 𝑐1𝑒− 1 2 log 𝑥 + 𝑐2𝑒 1 2 log 𝑥 ⟹ 𝑐1 𝑥 + 𝑐2 𝑥 , since 𝑒log 𝑥 = 𝑥
- 7. Problem: Solve 𝟒𝒙𝟐𝒅𝟐𝒚 𝒅𝒙𝟐 + 𝟒𝒙 𝒅𝒚 𝒅𝒙 − 𝒚 = 𝟒𝒙𝟐 (Cont…) Now, Particular integral, 𝑃. 𝐼 = 1 4𝐷1 2 − 1 4 𝑒2𝑧 ⟹ 𝑃. 𝐼 = 4 1 15 𝑒2𝑧 ⟹ 𝑃. 𝐼 = 4 15 𝑥2
- 8. Problem: Solve 𝟒𝒙𝟐𝒅𝟐𝒚 𝒅𝒙𝟐 + 𝟒𝒙 𝒅𝒚 𝒅𝒙 − 𝒚 = 𝟒𝒙𝟐 (Cont…) Hence the complete solution is 𝑦 = 𝐶. 𝐹 + 𝑃. 𝐼 ⟹ 𝑦 = 𝑐1 𝑥 + 𝑐2 𝑥 + 4 15 𝑥2