Question #193725

Find the partial differential equation whose general solution is given by the arbitrary

function F(2x + 3y + 4z , x^2 + y^2 − 2) = 0


1
Expert's answer
2021-05-18T17:28:56-0400

Given function is-


F(2x+3y+4z,x2+y22)=0F(2x+3y+4z,x^2+y^2-2)=0


Here, u=2x+3y+4z,v=x2+y22u=2x+3y+4z,v=x^2+y^2-2


Now differentiate f w.r.t x-


dfdu.dudx+dfdv.dvdx=0         (1)\dfrac{df}{du}.\dfrac{du}{dx}+\dfrac{df}{dv}.\dfrac{dv}{dx}=0~~~~~~~~~-(1)


Also differentiate w.r.t y-


dfdu.dudy+dfdv.dvdy=0         (2)\dfrac{df}{du}.\dfrac{du}{dy}+\dfrac{df}{dv}.\dfrac{dv}{dy}=0~~~~~~~~~-(2)


Now eleminating function f between eqn.(1) and (2) and we get the required PDE as-


dudxdvdxdudydvdy=0\begin{vmatrix}\dfrac{du}{dx}&\dfrac{dv}{dx}\\\\\dfrac{du}{dy}&\dfrac{dv}{dy}\end{vmatrix}=0


dudx2xdudy2y=0\begin{vmatrix} \dfrac{du}{dx}& 2x\\\\\dfrac{du}{dy}&2y\end{vmatrix}=0


2ydudx2xdudy=02y\dfrac{du}{dx}-2x\dfrac{du}{dy}=0


Hence The required PDE is 2ydudx2xdudy=02y\dfrac{du}{dx}-2x\dfrac{du}{dy}=0


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