Answer to Question #266764 in Differential Equations for Zuru

Question #266764

Solve: z(x+2y)p-z(2x+y)q=y^2-x^2


1
Expert's answer
2021-11-17T14:46:58-0500

Solution;

z(x+2y)pz(2x+y)q=y2x2z(x+2y)p-z(2x+y)q=y^2-x^2

The auxiliary equation are;

dxz(x+2y)=dyz(2x+y)=dzy2x2\frac{dx}{z(x+2y)}=\frac{dy}{-z(2x+y)}=\frac{dz}{y^2-x^2}

Considering the first two equations;

dxdy3z(x+y)=0\frac{dx-dy}{3z(x+y)}=0 .....(1)

dx=dydx=dy

Integrating;

x=y+c1x=y+c_1

Considering (1) and the last auxillary equation;

dxdy3z(x+y)=dzx2y2\frac{dx-dy}{3z(x+y)}=\frac{dz}{x^2-y^2}

(xy)(dxdy)=3zdz(x-y)(dx-dy)=3zdz

xdxxdyydx+ydy=3zdzxdx-xdy-ydx+ydy=3zdz

Integrating;

x222xy+y22=32z2+c2\frac{x^2}{2}-2xy+\frac{y^2}{2}=\frac 32z^2+c_2

x24xy+y2=3z2+c2x^2-4xy+y^2=3z^2+c_2

The solution of the equation is;

F(xy,x2+y23z24xy)=0F(x-y,x^2+y^2-3z^2-4xy)=0


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