🌈 Complete Solutions to Exercise 8.7 🌈
Problem 1: Determine Homogeneity
1(i)
f(x,y) = x²y + 6x³ + 7
1. Replace x→tx and y→ty:
f(tx,ty) = (tx)²(ty) + 6(tx)³ + 7 = t³x²y + 6t³x³ + 7
2. Check homogeneity:
The first two terms require t³, but the constant term requires t⁰.
3. Conclusion:
Not homogeneous because terms don't scale uniformly.
Final Answer: Not homogeneous ❌
1(ii)
h(x,y) = (6x²y³ - πy⁵ + 9x⁴y) / (2020x² + 2019y²)
1. Analyze numerator and denominator:
Numerator degree: 5 (all terms)
Denominator degree: 2 (both terms)
2. Apply scaling:
h(tx,ty) = [t⁵(6x²y³ - πy⁵ + 9x⁴y)] / [t²(2020x² + 2019y²)] = t³h(x,y)
Final Answer: Homogeneous of degree 3 ✅
1(iii)
g(x,y,z) = √(3x² + 5y² + z²) / (4x + 7y)
1. Analyze degrees:
Numerator: √(degree 2) → degree 1
Denominator: degree 1
2. Apply scaling:
g(tx,ty,tz) = [t√(3x²+5y²+z²)] / [t(4x+7y)] = t⁰g(x,y,z)
Final Answer: Homogeneous of degree 0 ✅
1(iv)
U(x,y,z) = xy + sin((y² - 2z²)/(xy))
1. Analyze terms:
First term (xy): degree 2
sin argument: (degree 2)/(degree 2) → degree 0
2. Apply scaling:
U(tx,ty,tz) = t²xy + sin((y²-2z²)/(xy))
The first term scales with t², but the sin term doesn't scale.
Final Answer: Not homogeneous ❌
Problem 2: Verify Homogeneity & Euler's Theorem
2
f(x,y) = x³ - 2x²y + 3xy² + y³
1. Check homogeneity:
f(tx,ty) = t³x³ - 2t³x²y + 3t³xy² + t³y³ = t³f(x,y)
Homogeneous of degree 3
2. Verify Euler's theorem:
Compute partial derivatives:
∂f/∂x = 3x² - 4xy + 3y²
∂f/∂y = -2x² + 6xy + 3y²
Compute x∂f/∂x + y∂f/∂y:
= x(3x²-4xy+3y²) + y(-2x²+6xy+3y²)
= 3x³ - 6x²y + 9xy² + 3y³ = 3f(x,y)
Final Answer: Homogeneous of degree 3 ✅
Euler's theorem verified!
Problem 3: Verify Homogeneity & Euler's Theorem
3
g(x,y) = x log(y/x)
1. Check homogeneity:
g(tx,ty) = tx log(ty/tx) = t[x log(y/x)] = t g(x,y)
Homogeneous of degree 1
2. Verify Euler's theorem:
Compute partial derivatives:
∂g/∂x = log(y/x) - 1
∂g/∂y = x/y
Compute x∂g/∂x + y∂g/∂y:
= x(log(y/x)-1) + y(x/y) = x log(y/x) = g(x,y)
Final Answer: Homogeneous of degree 1 ✅
Euler's theorem verified!
Problem 4: Prove the Given Relation
4
u(x,y) = (x² + y²)/√(x + y)
1. Check homogeneity:
u(tx,ty) = (t²x² + t²y²)/√(tx + ty) = t²(x²+y²)/[t⁰·⁵√(x+y)] = t¹·⁵ u(x,y)
Homogeneous of degree 1.5 (3/2)
2. Apply Euler's theorem:
For homogeneous functions:
x∂u/∂x + y∂u/∂y = (degree) × u
Here degree = 3/2, so:
x∂u/∂x + y∂u/∂y = (3/2)u
Final Answer: Relation proven! ✅
Problem 5: Prove the Given Relation
5
v(x,y) = log((x² + y²)/(x + y))
1. Check modified homogeneity:
v(tx,ty) = log((t²x² + t²y²)/(tx + ty)) = log(t) + v(x,y)
This is logarithmically homogeneous
2. Apply modified Euler's theorem:
For functions where f(tx,ty) = f(x,y) + k log(t):
x∂v/∂x + y∂v/∂y = k
Here k=1, so:
x∂v/∂x + y∂v/∂y = 1
Final Answer: Relation proven! ✅
Problem 6: Find the Given Expression
6
w(x,y,z) = log((5x³y⁴ + 7y²xz⁴ - 75y³z⁴)/(x² + y²))
1. Analyze degrees:
Numerator: 5x³y⁴ (degree 7), 7y²xz⁴ (degree 7), -75y³z⁴ (degree 7)
Denominator: x² + y² (degree 2)
2. Apply scaling:
w(tx,ty,tz) = log(t⁷N/t²D) = 5log(t) + w(x,y,z)
3. Apply modified Euler's theorem:
For w(tx,ty,tz) = w(x,y,z) + k log(t):
x∂w/∂x + y∂w/∂y + z∂w/∂z = k = 5
Final Answer: x∂w/∂x + y∂w/∂y + z∂w/∂z = 5 ✅