Answer to Question #193725 in Differential Equations for aparajita mishra

Question

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

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

Answer 1

Given function is-

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

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

Now differentiate f w.r.t x-

$\dfrac{df}{du}.\dfrac{du}{dx}+\dfrac{df}{dv}.\dfrac{dv}{dx}=0~~~~~~~~~-(1)$

Also differentiate w.r.t y-

$\dfrac{df}{du}.\dfrac{du}{dy}+\dfrac{df}{dv}.\dfrac{dv}{dy}=0~~~~~~~~~-(2)$

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

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

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

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

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