Question

A function f(x,y) is said to be homogeneous of degree m if f(λx,λy)=λ m f(x,y) for all values of λ for which (λx,λy) is in the domain of f. If f(x,y) is homogeneous of degree zero, then the differential equation u ′ (x)=f(x,y) is called a homogeneous first order different equation. A substitution u(x)=xv(x) where v(x) is the new dependent function will transform ahomogeneous first order different equation to a separable first order differential equation. A family of curves, parameterized by λ, in the xy-plane satisfies the following coupled differential equations: {dy/dx =-4x-y dx/dλ=2x-y Determine this family of curves which is known as integral curves.

Answer 1

A function f(x,y) is said to be homogeneous of degree m if f(λx,λy)=λmf(x,y) for all values of λ for which (λx,λy) is in the domain of f. If f(x,y) is homogeneous of degree zero, then the differential equation u ′ (x)=f(x,y) is called a homogeneous first order different equation. A substitution u(x)=xv(x) where v(x) is the new dependent function will transform a homogeneous first order different equation to a separable first order differential equation.

Step-by-step explanation:

A function f(x,y) is said to be homogeneous of degree m if f(λx,λy)=λmf(x,y) for all values of λ for which (λx,λy) is in the domain of f. If f(x,y) is homogeneous of degree zero, then the differential equation u ′ (x)=f(x,y) is called a homogeneous first order different equation. A substitution u(x)=xv(x) where v(x) is the new dependent function will transform a homogeneous first order different equation to a separable first order differential equation. A family of curves, parameterized by λ, in the xy-plane satisfies the following coupled differential equations: {dy/dx =-4x-y dx/dλ=2x-y Determine this family of curves which is known as integral curves.