Answer to Question #178192 in Differential Equations for m siva prasad

Question #178192

Solve yz2 (x2-yz)dx+zx2 (y2-zx)dy+xy2 (z2-xy)dz=0


1
Expert's answer
2021-04-15T06:57:31-0400

The condition of integrability:

P(QzRy)+Q(RxPz)+R(PyQx)=0P(\frac{\partial Q}{\partial z}-\frac{\partial R}{\partial y})+Q(\frac{\partial R}{\partial x}-\frac{\partial P}{\partial z})+R(\frac{\partial P}{\partial y}-\frac{\partial Q}{\partial x})=0

(yx2z2y2z3)(x2y22zx32yxz2+3y2x2)+(zx2y2z2x3)(y2z22xy3(yx^2z^2-y^2z^3)(x^2y^2-2zx^3-2yxz^2+3y^2x^2)+(zx^2y^2-z^2x^3)(y^2z^2-2xy^3-

2zyx2+3z2y2)+(xy2z2x2y3)(x2z22yz32xzy2+3x2z2)=-2zyx^2+3z^2y^2)+(xy^2z^2-x^2y^3)(x^2z^2-2yz^3-2xzy^2+3x^2z^2)=

=y3x4z22yx5z32y2x3z4+3y3x4z2x2y4z3+2x3y2z4+2xy3z53x2y4z3+=y^3x^4z^2-2yx^5z^3-2y^2x^3z^4+3y^3x^4z^2-x^2y^4z^3+2x^3y^2z^4+2xy^3z^5-3x^2y^4z^3+

+x2y4z32x3y5z2x4y3z2+3x2y4z3x3y2z4+2x4y3z2+2x5yz33x3y2z4++x^2y^4z^3-2x^3y^5z-2x^4y^3z^2+3x^2y^4z^3-x^3y^2z^4+2x^4y^3z^2+2x^5yz^3-3x^3y^2z^4+

+x3y2z42xy3z52x2y4z3+3x3y2z4x4y3z2+2x2y4z3+2x3y5z3x4y3z2=+x^3y^2z^4-2xy^3z^5-2x^2y^4z^3+3x^3y^2z^4-x^4y^3z^2+2x^2y^4z^3+2x^3y^5z-3x^4y^3z^2=

=0=0

The equation is integrable.

The equation is homogeneous.

D=Px+Qy+Rz=xyz2(x2yz)+yzx2(y2zx)+zxy2(z2xy)=D=Px+Qy+Rz=xyz^2(x^2-yz)+yzx^2(y^2-zx)+zxy^2(z^2-xy)=

=x3yz2xy2z3+x2y3zx3yz2+xy2z3x2y3z=0=x^3yz^2-xy^2z^3+x^2y^3z-x^3yz^2+xy^2z^3-x^2y^3z=0


Let

x=uz,y=vzx=uz,y=vz

dx=udz+zdu,dy=vdz+zdvdx=udz+zdu,dy=vdz+zdv

Then:

vz3(u2z2vz2)(udz+zdu)+z3u2(v2z2z2u)(vdz+zdv)+vz^3(u^2z^2-vz^2)(udz+zdu)+z^3u^2 (v^2z^2-z^2u)(vdz+zdv)+

+uv2z3(z2uvz2)dz=0+uv^2z^3(z^2-uvz^2)dz=0

(vu3v2u)dz+(vu2zv2z)du+(v3u2vu3)dz+(zv2u2zu3)dv+(vu^3-v^2u)dz+(vu^2z-v^2z)du+(v^3u^2-vu^3)dz+(zv^2u^2-zu^3)dv+

+(uv2u2v3)dz=0+(uv^2-u^2v^3)dz=0

vz(u2v)du+u2z(v2u)dv=0vz(u^2-v)du+u^2z(v^2-u)dv=0

v(u2v)du+u2(v2u)dv=0v(u^2-v)du+u^2(v^2-u)dv=0

dvdu=v(u2v)u2(uv2)\frac{dv}{du}=\frac{v(u^2-v)}{u^2(u-v^2)}


v=(c+1)24u3+cu+12uv=\frac{\sqrt{(c+1)^2-4u^3}+cu+1}{2u}


yz=z((c+1)24(x/z)3+(cx/z)+1)2x\frac{y}{z}=\frac{z(\sqrt{(c+1)^2-4(x/z)^3}+(cx/z)+1)}{2x}


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