Question

Show that the given differential equation is homogeneous and solve:$y'=\frac{x+y}{x}$

Answer 1

The given differential equation is:
$y'=\frac{x+y}{x}$
$\frac{dy}{sx}=\frac{x+y}{x}$
Let $F(x,y)=\frac{x+y}{x}.$
Now,$F(λx,λy)=\frac{λx+λy}{λx}=\frac{x+y}{x}=λ=F(x,y)$
Thus the given equation is a homogenous equation.
To solve it,we make the substitution as:
$y=vx$
Differentiating both sides with respect to x,we get:
$\frac{dy}{dx}=v+\frac{xdv}{dx}$
Substituting the values of y and dy/dx in equation(1),we get:
$v+x \frac{dx}{dy}=x+\frac{vx}{x}$
$⇒v+x\frac{dv}{dx}=1+v$
$x\frac{dv}{dx}=1$
$⇒dv=\frac{dx}{x}$
Integrating both sides,we get:
$v=logx+C$
$⇒\frac{y}{x}=logx+C$
$⇒y=xlogx+Cx$
This is the required solution of the given differential equation.