$(03)($a$)$ Evaluate

$($i$)$ the auxiliary equation,

$($ii$)$ complementary function,

$($iii$)$ particular integral, and the

$($iv$)$ general solution,

of the following differential equations.

$($A$)(D^2+4D+4)y=e^{-3x}$

$06$ Marks

$($B$)(D^2-3D+2)y=sin3x$

$06$ Marks

$($b$)$ Find all first order and second order partial derivatives of the function

$f(x,y)=e^{2x}{y}^2+xsin(2xy)$

Hence show that $\frac{de{l}^{2}f}{\text{d}\text{e}\text{l}\text{x}\text{d}\text{e}\text{l}\text{y}}=\frac{de{l}^{2}f}{\text{d}\text{e}\text{l}\text{y}\text{d}\text{e}\text{l}\text{x}}.$

$07$ Marks

$($c$)$ Show that ${}_R(xy-y^3)dxdy=-\frac{23}{40},$ where $R$ is the region consisting the square $\{(x,y)$:$-1x0,0y1\}$ together with the triangle $\{(x,y)$:$xy1,0x1\}.$

$06$ Marks