Q1:Consider equations and determine their order; determine if they are linear homogeneous, linear inhomogeneous or non-linear (𝑢 is an unknown function):
$u_{tt}+u_{xxxx}+u=0$
Ans:Since the highest order derivative from the left part is $u_{xxxx}$ and its order is 4,
the whole equation has order 4. It is
linear homogeneous, since all terms have degree of 1 and are related to u.
Q2:Find the general solutions to the following equations:
$u_{xxy}=sin(x)sin(y)$
Ans:$u_{xx}=\int {sin(x)sin(y)}dy$
$u_{xx}=-sin(x)cos(y)+\varphi _1(x)$
$u_{x}=+cos(x)cos(y)+\varphi _2(x)+\psi _1(y)$
$u=sin(x)cos(y)+\varphi _3(x)+x\psi _1(y)+\psi _2(y)$
