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߲ ݑ ߲ݕ ߲ ݑ ߲ݖ ʹ ݔ ൅ ݕ ൅ ݖ 4. ࢝ ൌ ࢌሺ࢟ െ ࢠǡ ࢠ െ ࢞ǡ ࢞ െ ࢟ሻ show that ࣔ࢝ ࣔ࢞ ࣔ࢝ ࣔ࢟ ࣔ࢝ ࣔࢠ ൌ ૙ Solution: Let ݓ ൌ ݂ሺݎǡ ݏǡ ݐሻ , ݎ ൌ ݕ െ ݖǡ ݏ ൌ ݖ െ ݔǡ ݐ ൌ ݔ െ ݕ డ௪ డ௫ డ௪ డ௥ డ௥ డ௫ డ௪ డ௦ డ௦ డ௫ డ௪ డ௧ డ௧ డ௫ డ௪ డ௦ ሺെͳሻ ൅ డ௪ డ௧ ሺͳሻ ---------------- (1) డ௪ డ௬ డ௪ డ௥ డ௥ డ௬ డ௪ డ௦ డ௦ డ௬ డ௪ డ௧ డ௧ డ௬ డ௪ డ௥ ሺͳሻ ൅ డ௪ డ௧ ሺെͳሻ ---------------- (2) డ௪ డ௭ డ௪ డ௥ డ௥ డ௭ డ௪ డ௦ డ௦ డ௭ డ௪ డ௧ డ௧ డ௭ డ௪ డ௥ ሺെͳሻ ൅ డ௪ డ௦ ሺͳሻ ---------------- (3) (1)+ (2)+ (3) ൌ൐ డ௪ డ௫ డ௪ డ௬ డ௪ డ௭ ൌ Ͳ 5. If ࢛ ൌ ܜ܉ܖ ି૚ ା࢟ ࢞ି࢟ , Show that ࢛ࣔ ࣔ࢞ ൅ ࢟ ࢛ࣔ ࣔ࢟ ൌ ܛܑܖ ૛࢛ Solution: Given ݖ ൌ ݑ ൌ ା௬ ௫ି௬ z is a homogeneous function of degree n=2 ߲ݖ ߲ݔ ݑ ߲ݑ ߲ݔ ߲ݖ ߲ݕ ݑ ߲ݑ ߲ݕ As z is homogeneous function of order n=2, it satisfies the Eulers equation ݔ ߲ݖ ߲ݔ ൅ ݕ ߲ݖ ߲ݕ ൌ ʹݖ ݑ ߲ݑ ߲ݔ ൅ ݕ ݑ ߲ݑ ߲ݕ ൌ ʹ ݑ ݔ ߲ݑ ߲ݔ ൅ ݕ ߲ݑ ߲ݕ ʹݑ
83 6. If u = , find ࣔ࢞ࣔ࢟ Sol: given u = ݔ ൌ ݁ ௬௟௢௚௫ ߲ݑ ߲ݕ ൌ ݁ ௬௟௢௚௫ ߲ ߲ݕ ሺݕ݈݋݃ݔሻ ൌ ݈݋݃ݔ ݁ ௬௟௢௚௫ ߲ ݑ ߲ݔ߲ݕ ߲ ߲ݔ ߲ݑ ߲ݕ ൰ ൌ ߲ ߲ݔ ൫݈݋݃ݔ ݁ ௬௟௢௚௫ ൯ ൌ ݈݋݃ݔ ݁ ௬௟௢௚௫ ߲ ߲ݔ ሺݕ݈݋݃ݔሻ ൅ ݁ ௬௟௢௚௫ ͳ ݔ ߲ ݑ ߲ݔ߲ݕ ൌ ݈݋݃ݔሺݔ ሻ ቀ ݕ ݔ ቁ ൅ ݔ ݔ ൌ ݔ ௬ିଵ ሺݕ݈݋݃ݔ ൅ ͳሻ Euler`s Theorem for Homogeneous Function Euler`s Theorem: If u be a homogeneous function of degree n an x and y then ࢞ ቀ ࢛ࣔ ࣔ࢞ ቁ ൅ ࢟ ቀ ࢛ࣔ ࣔ࢟ ቁ ൌ ࢔࢛ Problems: 1. Verify Eulers theorem for the function ࢛ ൌ ࢞ ൅ ࢟ ൅ ૛࢞࢟ Solution: Given ݑ ൌ ݔ ൅ ݕ ൅ ʹݔݕ This is a homogeneous function of degree 2. ߲ݑ ߲ݔ ൌ ʹݔ ൅ ʹݕ ݔ ߲ݑ ߲ݔ ൌ ʹݔ ൅ ʹݔݕ ǥ ǥ ǥ ǥ ǥ ǥ ǥ ǥ Ǥ ሺͳሻ ݕ ߲ݑ ߲ݔ ൌ ʹݕ ൅ ʹݔݕ ǥ ǥ ǥ ǥ ǥ ǥ ǥ ǥ ǥ ሺʹሻ Adding (1) and (2) we get ݔ డ௨ డ௫ ൅ ݕ డ௨ డ௫ ൌ ʹݑ Hence Eulers theorem is verified. 2. If ࢛ ൌ ܛܑܖ ି૚ ඥ࢞ ൅ ࢟ , Show that ࣔ࢞ ൅ ૛࢞࢟ ࣔ࢞ࣔ࢟ ൅ ࢟ ࣔ࢟ ൌ ܜ܉ܖ Solution: Given ݖ ൌ ݑ ൌ ඥݔ ൅ ݕ As z is a homogeneous function of order n = 1, it satisfies the Eulers theorem ݔ డ௭ డ௫ ൅ ݕ డ௭ డ௬ ൌ ͳ ݖ ൌ ݑ ---------------- (1) ߲ݖ ߲ݔ ݀ݖ ݀ݑ ߲ݑ ߲ݔ ݑ ߲ݑ ߲ݔ ߲ݖ ߲ݕ ݀ݖ ݀ݑ ߲ݑ ߲ݕ ݑ ߲ݑ ߲ݕ (1) ൌ൐ ݔ డ௭ డ௫ ൅ ݕ డ௭ డ௬ ݑ ---------------- (2) By Eulers theorem of second order ݔ డ௫ ൅ ʹݔݕ డ௫డ௬ ൅ ݕ డ௬ ൌ ݂ሺݑሻ൫݂ ሺݑሻ െ ͳ൯ ---------------- (3) ݂ሺݑሻ ൌ ݑ , ݂ ሺݑሻ ൌ ݑ put in (3)
84 ݔ ߲ ݑ ߲ݔ ൅ ʹݔݕ ߲ ݑ ߲ݔ߲ݕ ൅ ݕ ߲ ݑ ߲ݕ ݑ ሺ ݑ െ ͳሻ ൌ ݑ 3. If u = ܛܑܖ ି૚ ξ࢞ ିඥ࢟ ξ࢞ ାඥ࢟ , prove that ࢛ࣔ ࣔ࢞ ൅ ࢟ ࢛ࣔ ࣔ࢟ ൌ ૙ Sol: given ݏ݅݊ݑ ൌ ξ௫ ି ξ ξ௫ ξ ξ௫ ൬ଵି ඥ೤ ξೣ ξ௫ ൬ଵା ඥ೤ ξೣ = ݔ ߮ ቀ ׵ ݏ݅݊ݑ is a homogeneous function of degree 0. Hence by Euler’s theorem, we have ݔ డ௨ డ௫ ൅ ݕ డ௨ డ௬ ൌ Ͳ x u = 0 Total Derivatives-Differentiation of Implicit Function Total Derivative: If ݑ ൌ ݂ሺݔǡ ݕሻǡ ݓ݄݁ݎ݁ ݔ ൌ ߶ሺݐሻ ܽ݊݀ ݕ ൌ ߰ሺݐሻ , then we can express u as a function of t alone by substituting the value of x and y in ݂ሺݔǡ ݕሻ Thus we can find the ordinary derivatives ௗ௨ ௗ௧ which is called the total derivative of u distinguish it from partial derivatives డ௨ డ௫ ܽ݊݀ డ௨ డ௬ Now to find the ௗ௨ ௗ௧ without actually substituting the values of x and y in ݂ሺݔǡ ݕሻ we establish the following formula ࢊ࢛ ࢊ࢚ ࢛ࣔ ࣔ࢞ ࢊ࢞ ࢊ࢚ ࢛ࣔ ࣔ࢟ ࢊ࢟ ࢊ࢚ Problems: 1. Find ࢊ࢛ ࢊ࢚ if ࢛ ൌ ࢞ where ࢞ ൌ ࢚ and ࢟ ൌ ࢚ Solution: ௗ௨ ௗ௧ డ௨ డ௫ ௗ௫ ௗ௧ డ௨ డ௬ ௗ௬ ௗ௧ ൌ ͵ݔ ݕ ͵ݐ ൅ Ͷݔ ݕ ʹݐ ൌ ͳ͹ݐ ଵ଺ 2. If ࢛ ൌ ࢋ ܛܑܖ ࢟ where ࢞ ൌ ࢙࢚ and ࢟ ൌ ࢙ find ࢛ࣔ ࢙ࣔ and ࢛ࣔ ࢚ࣔ Solution:

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