391 | If log u = (x³+y³)/(3x+4y) Prove x∂u/∂x + y∂u/∂y = 2u log u | Euler's Theorem | GATE Maths

391 | If log u = (x³+y³)/(3x+4y) Prove x∂u/∂x + y∂u/∂y = 2u log u | Euler's Theorem | GATE Maths Partial DIFFERENTIATION playlist -    • PARTIAL DIFFERENTIATION   Notes-    • 391 | If log u = (x³+y³)/(3x+4y) Prove x∂u...   Description: Prove x(∂u/∂x)+y(∂u/∂y) = 2u log u for log u = (x³+y³)/(3x+4y) using Euler's Theorem for Homogeneous Functions! 🚀 Tricky GATE, JEE Advanced, BTech, BSc Engineering Mathematics problem: In this video we'll learn: 00:00 If logu = (x³+y³)/(3x+4y) then show that x(∂u/∂x)+y(∂u/∂y) = 2ulogu 01:00 Euler's Theorem for Homogeneous Function 03:00 x(∂u/∂x)+y(∂u/∂y) Perfect for JEE Main, Advanced, GATE, ESE, BTech, BSc, and MSc students. Solution: Given log u = (x³+y³)/(3x+4y). Let f = log u. Step 1: Check degree of f f(tx,ty) = (t³x³+t³y³)/(3tx+4ty) = t³(x³+y³)/t(3x+4y) = t²f. So f = log u is homogeneous of degree n = 2. Step 2: Apply Euler's Theorem on f x_∂f/∂x + y_∂f/∂y = 2f = 2 log u. Step 3: Convert to u f = log u, so ∂f/∂x = (1/u)_∂u/∂x = Ux/u. Thus x_(Ux/u) + y_(Uy/u) = 2 log u (xUx + yUy)/u = 2 log u xUx + yUy = 2u log u. Hence proved. Key trick: When question gives log u = homogeneous function, apply Euler's Theorem directly on log u, not on u. If u = f and f is homogeneous of degree n, then xUx + yUy = nu. But here log u is homogeneous, so xUx + yUy = 2u log u. This pattern appears in GATE Mathematics and JEE Advanced. Other variations: u = e^f, u = sin f, u = tan^(-1)f where f is homogeneous. Subscribe for more Euler's Theorem, Homogeneous Functions, Logarithmic Differentiation, Partial Differentiation, and Engineering Mathematics! 📚💻 Like, share, and comment with your doubts! Some key topics covered in this video include: Euler's Theorem Homogeneous Functions Degree 2 Logarithmic Differentiation Partial Differentiation Engineering Mathematics 1 Multivariable Calculus GATE Mathematics PYQ Type JEE Advanced Calculus xUx + yUy = 2u log u Chain Rule in PDE Hashtags: #EulersTheorem #HomogeneousFunction #PartialDerivatives #Calculus #EngineeringMathematics #Logarithm #MathShorts #GATE #JEEAdvanced #BTech #BSc #MSc #MathExplained #MathTutorial #MultivariableCalculus _Most searching keywords:* Euler's Theorem, Homogeneous Function, Degree of homogeneity, x∂u/∂x+y∂u/∂y = 2u log u, log u = (x³+y³)/(3x+4y), Partial Derivatives, Calculus, Multivariable Calculus, Engineering Mathematics, GATE, JEE Advanced, BTech, BSc, MSc, Logarithmic differentiation, Homogeneity test, Euler's theorem formula, Math proof, log u homogeneous, Chain rule partial derivative