Question

Homogeneous Differential Equations of Differential Equations

Answer 1

Any function f(x, y) is said to be homogeneous function of power n, if f(λx, λy) = λⁿ f(x, y), where X is a non-zero constant.

For example,

For f(x, y) = x³ + y³ + x²y + xy²

f(λx, λy) = (λx)³ + {λy)³ + (λx)²(λy) + (λx)(λy)²

= λ³x³ + λ³y³ + λ³x²y + λ³xy²

= λ³ (x³ + y³ + x²y + xy²)

= λ³f(x,y)

Hence, function f(x, y) is a homogeneous function of degree 3.

Similarly, F(x,y) = 3x + 4y

And H(x, y) = tan \((\frac{x}{y})\)

For F(λx, λy) = 3λx + 4(λy)

= λ(3x + 4y) = λ ≠ F(x, y)

Hence, F(x, y) is a homogeneous function of degree 1.

And H(λx, λy) = tan \((\frac{\lambda x}{\lambda y})\) = tan \((\frac{ x}{ y})\)

= λ° tan \((\frac{ x}{ y})\) (∵ λ° = 1)

Hence, H(x, y) is a homogenerous function of degree 0.

Now, for the function G(x, y) = sec x + tan y

G(λx, λy) = sec (λx) + tan (λy)

≠ λ(sec x + tan y)

∴ G(λx, λy) ≠ λG(x, y)

Hence, function G(x, y), is not homogeneous function.

Hence, we can write a homogeneous function of degree in the following way :

f(x, y) = xⁿ f \((\frac{ y}{ x})\) 1 or f(x, y) = yⁿ \((\frac{ x}{ y})\)