Answer to Question #188636 in Differential Equations for mahammad

Question #188636
erify that the following cquations are integrable and determine their solutions.
x(y^2-a^2 )dx+y(x^2-t^2 )dy-z(y^2-α^2 )dz=0, where α is constant
1
Expert's answer
2021-05-07T11:36:42-0400

The given equation is-

x(y2a2)dx+y(x2t2)dyz(y2α2)dz=0,x(y^2-a^2 )dx+y(x^2-t^2 )dy-z(y^2-α^2 )dz=0,


Given equation is of the form-

Pp+Qq+Rr=0Pp+Qq+Rr=0


The dxP=dyQ=dzR\dfrac{dx}{P}=\dfrac{dy}{Q}=\dfrac{dz}{R}


So,dxx(y2a2)=dyy(x2t2)=dzz(y2α2)\dfrac{dx}{x(y^2-a^2)}=\dfrac{dy}{y(x^2-t^2)}=\dfrac{dz}{z(y^2-\alpha^2)}



Taking first two terms-

 dxx(y2a2)=dyy(x2t2)\dfrac{dx}{x(y^2-a^2)}=\dfrac{dy}{y(x^2-t^2)}



(x2t2)dxx=(y2a2)dyy\Rightarrow \dfrac{(x^2-t^2)dx}{x}=\dfrac{(y^2-a^2)dy}{y}


Integrating both the sides-


x22t2lnx=y22a2lny+c1\dfrac{x^2}{2}-t^2lnx=\dfrac{y^2}{2}-a^2lny+c_1


x22t2lnxy22+a2lny=c1         (1)\Rightarrow \dfrac{x^2}{2}-t^2lnx-\dfrac{y^2}{2}+a^2lny=c_1~~~~~~~~~-(1)


Taing last two terms-


dyy(x2t2)=dzz(y2α2)\dfrac{dy}{y(x^2-t^2)}=\dfrac{dz}{z(y^2-\alpha^2)}



(y2α2)dyy=(z2t2)dzz\Rightarrow \dfrac{(y^2-\alpha^2)dy}{y}=\dfrac{(z^2-t^2)dz}{z}


Integrating Both the sides-


y22α2lny=z22t2lnz+c2\dfrac{y^2}{2}-\alpha^2lny=\dfrac{z^2}{2}-t^2lnz+c_2


y22α2lnyz22+t2lnz=c2\Rightarrow \dfrac{y^2}{2}-\alpha^2lny-\dfrac{z^2}{2}+t^2lnz=c_2



Hence The solution is-


ϕ(c1,c2)=0\phi(c_1,c_2)=0


ϕ(x22t2lnxy22+a2lny,y22α2lnyz22+t2lnz)=0\phi (\dfrac{x^2}{2}-t^2lnx-\dfrac{y^2}{2}+a^2lny, \dfrac{y^2}{2}-\alpha^2lny-\dfrac{z^2}{2}+t^2lnz)=0




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