Answer to Question #187475 in Differential Equations for imran

Question

Determine ‘a’ and ‘b’ as to make the cylinder y2= 4ax orthogonal to the ellipsoid at the point (1, 2, 1).
Find the directional derivative of f(x,y)=2x^2+y^2 at the point (-2,3) in the direction of u ⃗=4i ̂-5j ̂.

Show that the function f(x,y)=e^x siny+e^y cosx satisfies the Laplace’s equation

Answer 1

Ans:-

$f(x,y)=e^x siny+e^y cosx$

partial derivative with respect to x partial derivative with respect to y

$\frac{\partial f}{\partial x} =e^x siny-e^ysinx$ , $\frac{\partial f}{\partial y}=e^xcosy+e^ycosx$

again partial derivative with respect to x again partial derivative with respect to y

$\frac{\partial^2 f}{\partial x^2}=e^xsiny-e^ycosx$ $-(i)$ $\frac{\partial^2 f}{\partial x^2}=-e^xsiny+e^ycosx$ $-(ii)$

Add these two equations

$\Rightarrow \frac{\partial^2 f}{\partial x^2} +\frac{\partial^2 f}{\partial y^2}=e^xsiny-e^ycosx+(-e^xsiny+e^ycosx)$ $=0$

$\Rightarrow \frac{\partial^2 f}{\partial x^2} +\frac{\partial^2 f}{\partial y^2}=0$

Hence Laplace's equation will be satisfied.