If $u \equiv u(x, y)=\sin (y+a x)-(y+a x)^2$, then it implies

If $u \equiv u(x, y)=\sin (y+a x)-(y+a x)^2$, then it implies
  1. $u_{x x}=a^2 \cdot u_{y y}$
  2. $u_{y y}=a^2 u_{x x}$
  3. $u_{x x}=-a^2 \cdot u_{y y}$
  4. $u_{y y}=-a^2 u_{x x}$

Solution


On differentiating partially w.r.t. $x$, we get $u_x=\cos (y+a x) a-2(y+a x) a$ Again differentiating partially w.r.t. $x$, we get
On differentiating partially Eq. (i) w.r.t. $y$, we get $u_y=\cos (y+a x)-2(y+a x)$
$\therefore$ From Eqs. (ii) and (iii), we get $u_{x x}=a^2 u_{y y}$

Asked in: AP EAMCET 2011

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