I was thinking about the following problem :
Find out which of the following option(s) is/are correct?
The differential equation $xu_{x}+yu_{y}=2u$ satisfying the initial condition $y=xg(x),u=f(x)$, with
(a)$f(x)=2x,g(x)=1$ has no solution,
(b)$f(x)=2x^2,g(x)=1$ has infinite number of solutions,
(c)$f(x)=x^3,g(x)=x$ has a unique solution,
(d)$f(x)=x^4,g(x)=x$ has a unique solution.
My Attempt:
Using lagrange's method, we see that $\frac{dx}{x}=\frac{dy}{y}=\frac{du}{2u} $ gives $y/x=c_1,$ and $y=\sqrt uc_2$ ,where $c_1,c_2$ being constants. Now we have to apply to the given initial conditions and check the given options.Here ,I observe that $y=xg(x)=x.1=x$,[as $g(x)=1$] and $u=f(x)=2x^2$But,this relation does not satisfy $xu_{x}+yu_{y}=2u$ and so the choice"(a)$f(x)=2x,g(x)=1$ has no solution," is right. Now I am stuck here and could not progress further.Am i going in the right direction? Please help.Thanks in advance for your time.