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:**

$u_{xx}=\int {sin(x)sin(y)}dy$

$u_{xx}=-sin(x)cos(y)+\varphi _1(x)$

$u_{x}=+cos(x)cos(y)+\varphi _2(x)+\psi _1(y)$

$u=sin(x)cos(y)+\varphi _3(x)+x\psi _1(y)+\psi _2(y)$