[Partial Differential Equation]
Show that each of the following equations has a solution of the form $ u(x,y)=f(ax+by) $ for a proper choice of constant $ a,b$ .

(a) $u_{x}+3u_{y}=0$
(b) $3u_{x}−7u_{y}=0$
(c)$ 2u_{x}+πu_{y}=0$

Question

[Partial Differential Equation] 
Show that each of the following equations has a solution of the form $ u(x,y)=f(ax+by) $ for a proper choice of constant $ a,b$ .

(a) $u_{x}+3u_{y}=0$
(b) $3u_{x}−7u_{y}=0$
(c)$ 2u_{x}+πu_{y}=0$

anonymous · Answer

ok this is a matter of substitution

anonymous · Answer

solving for part (a)$$a\frac{ \delta }{ \delta x }(ax+by)+3b\frac{ \delta }{ \delta y }(ax+by)=0$$

anonymous · Answer

or$$af \prime + 3b f \prime =0$$

anonymous · Answer

$$\ \therefore a=-3b$$

chihiroasleaf · Answer

here.., we conclude $a = -3b$ since by taking this value, the equation will become true and the function $f(ax+by)$ is the solution of the equation, right?

anonymous · Answer

and that satisfies the solution, so you try part b, and c, using the same method applied above

anonymous · Answer

yes

anonymous · Answer

i hope this helps

chihiroasleaf · Answer

part (b) $3a-7b =0$ 
(c) $2a+\pi b=0 $

the reason will be the same as part (a), 
right?

anonymous · Answer

yes

chihiroasleaf · Answer

ok..., thanks a lot... :)

anonymous · Answer

you're welcome