Question

The differential equation $x\frac{dy}{dx}=5y(\ln x-\ln y)$ is:

• A. a homogeneous but not linear differential equation.
• B. a linear but not homogeneous differential equation.
• C. both homogenous and linear differential equation.
• D. neither homogenous nor linear differential equation.

Answer 1

The correct option is A a homogeneous but not linear differential equation.

Given,Differential equation as $x\frac{dy}{dx}=5y\times ln\left(\frac{x}{y}\right)$

$\frac{dy}{dx}=5\left(\frac{y}{x}\right)ln\left(\frac{x}{y}\right)$

Let $f(x,y)=\frac{dy}{dx}$

$\Rightarrow f(x,y)=5\left(\frac{y}{x}\right)ln\left(\frac{x}{y}\right)$

For $f(x,y)$ to be homogenous,$f(kx,ky)=k^{n}f(x,y)$ (where n is its order)

Consider,$f(kx,ky)=5\left(\frac{ky}{kx}\right)ln\left(\frac{kx}{ky}\right)=5\left(\frac{y}{x}\right)ln\left(\frac{x}{y}\right)=f(x,y)$

$\therefore$The given equation is homogenous equation.

Consider given Differential equation, $x\frac{dy}{dx}=5y\times (lnx-lny)$

$\Rightarrow x\frac{dy}{dx}=5y\times lnx-5y\times lny$

For equation to be linear,

a)The powers of dependent variable and its derivative to be power of one.

b)The coefficient of dependent variable should be either function of independent variable or constant.

Since in our equation the co-efficient (for the term -5ylny) term is not function of x.The given equation is non-linear.

$\therefore Thegivenequationishomogeneousbutnon-linear$