Answer to Question #299382 in Differential Equations for Pankaj

Question #299382

Find the integral surface of the PDE:


x²p + y²q +z² =0


which passes through the hyperbola


xy =x+y, z=1.

1
Expert's answer
2022-02-22T08:36:28-0500

The auxiliary equation is dxx2=dyy2=dzz2.\dfrac{dx}{x^2}=\dfrac{dy}{y^2}=\dfrac{dz}{-z^2}.

Taking the first two ratios,


dxx2=dyy2Integrating both sides, we get1x=1y+c11y1x=c1(1)\begin{aligned} \dfrac{dx}{x^2}&=\dfrac{dy}{y^2}\\ \text{Integrating both sides, we get}\\ \dfrac{-1}{x} &= \dfrac{-1}{y} + c_{1}\\ \dfrac{1}{y} -\dfrac{1}{x} &= c_{1}\qquad\qquad(1)\\ \end{aligned}

Taking last two ratios,


dyy2=dzz2Integrating both sides, we get1y=1z+c21y1z=c2(2)\begin{aligned} \dfrac{dy}{y^2}&=-\dfrac{dz}{z^2}\\ \text{Integrating both sides, we get}\\ \dfrac{-1}{y} &= \dfrac{1}{z} + c_{2}\\ -\dfrac{1}{y} -\dfrac{1}{z} &= c_{2}\qquad\qquad(2)\\ \end{aligned}


The integral curve is ϕ(c1,c2)=0\phi(c_1,c_2) = 0, i.e., ϕ(1y1x,1y1z)=0\phi(\frac{1}{y}-\frac{1}{x},-\frac{1}{y}-\frac{1}{z}) = 0.


Given that the integral surface passes through xy=x+y,z=1xy=x+y, z=1.


Letting x(s)=sx(s) = s as the parametric representation, we get

sy=s+ysyy=sy(s1)=sy(s)=ss1\begin{aligned} sy &= s+y\\ sy-y&=s\\ y(s-1) &= s\\ \therefore y(s) &= \dfrac{s}{s-1} \end{aligned}

Using x(s)=s,y(s)=ss1,z=1x(s) = s, y(s) = \dfrac{s}{s-1}, z=1 in (1) and (2) we get

s1s1s=c1c1=s2sc1=12s1s=1c12(3)(s1s)1=c2c2=12ssc2=1s2(4)\begin{aligned} \frac{s-1}{s}-\frac{1}{s} &= c_1 \Rightarrow c_1 = \dfrac{s-2}{s}\Rightarrow c_1 = 1- \dfrac{2}{s} \Rightarrow \dfrac{1}{s} = \dfrac{1-c_1}{2}\qquad (3)\\ -\left(\frac{s-1}{s}\right)-1 &= c_2\Rightarrow c_2 = \dfrac{1-2s}{s} \Rightarrow c_2 = \dfrac{1}{s}-2 \qquad\qquad\qquad\qquad(4)\\ \end{aligned}


Using (3) in (4),

c2=1c1222c21+c1+4=0c1+2c2+3=0(1y1x)2(1y+1z)+3=0   (Using (1) & (2))xyxy2(z+yyz)+3=0xzyz2xz2xy+3xyz=03xyz2xyxzyz=0\begin{aligned} c_2 &= \dfrac{1-c_1}{2} - 2\\ 2c_2 - 1 + c_1+4 &= 0\\ c_1 + 2c_2+3 &=0\\ \left(\dfrac{1}{y}-\dfrac{1}{x}\right) - 2\left(\dfrac{1}{y}+\dfrac{1}{z}\right)+3&=0~~~(\text{Using (1) \& (2)})\\ \dfrac{x-y}{xy}-2\left(\dfrac{z+y}{yz}\right)+3&=0\\ xz-yz-2xz-2xy+3xyz &=0\\ 3xyz-2xy-xz-yz&=0 \end{aligned}

which is the required integral surface.


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