Home > Analysis, Olympiad, Partial Differential Equations > Traian Lalescu Student Contest 2011 Problem 4

## Traian Lalescu Student Contest 2011 Problem 4

Let $D=(0,\infty)\times (0,\infty),\ u \in C^1(D)$ and $\varepsilon>0$ fixed.

1) Prove that $x \frac{\partial u}{\partial x}(x,y)+y\frac{\partial u}{\partial y}(x,y)=u(x,y),\ \forall (x,y) \in D$ if and only if there exists $\phi \in C^1(0,\infty)$ such that $u(x,y)=x\phi(y/x),\ \forall (x,y) \in D$.

2) Prove that $\left|x \frac{\partial u}{\partial x}(x,y)+y\frac{\partial u}{\partial y}(x,y)-u(x,y)\right|\leq \varepsilon,\ \forall (x,y) \in D$ then there is a unique function $\phi \in C^1(0,\infty)$ such that $\left|u(x,y)-x\phi(y/x),\right| \leq \varepsilon \ \forall (x,y) \in D$.