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.