If u=f(r) , where r^2=x^2+y^2 , then ( ^2 u x^2 + ^2 u y^2 ) is equal to

Step-by-step solution

Given, u=f(r) and r^2=x^2+y^2 aligned & u_x=f^ (r) r x =f^ (r)_r^x & u_ x x =f^*(r) x^2 r^2 +f^ (r) r^2-x x r^2 & =f(r) x^2 r^2 +f^ (r) r^2-x^2 r^3 & u_y=f^ (r) r y =f^ (r) (2 y) 2 x^2+y^2 & =f^ (r) ( y r ) & and u_ y y =f^ (r) y^2 r^2 +f^r(r) r^2-y^2 r^3 & u_ x r +u_ y y =f^ (r) ( x^2 r^2 + y^2 r^2 ) & +f^ (r) ( r^2-x^2 r^3 + r^2-y^2 r^3 ) & =f^ (r) ( r^2 r^2 )+f^ (r) 2 r^2- (x^2+y^2 ) r^3 & =f^ (r)+f^ (r) r^2 r^3 & =f^ (r)+ f^ (r) r & aligned