I have to prove that the existence and uniqueness theorem for the solutions of a differential equation does not hold in the case of $t \frac{dx}{dt} =x$.
I guess this differential equation is equivalent to $\frac{dt} {t} =\frac{dx}{x}. $ I conclude that $x=t. $ In the problem there are not given initial conditions.
Can somebody explain why is this wrong, i. e. why for this equation the existence and uniqueness theorem does not hold.
Many thanks.
Best Answer
The general solution is $x=ct$ where $c$ is a constant. (You forgot the constant of integration). Hence there is no solution for the IVP $t\frac {dx} {dt} =x, x(0)=1$.