Question #93445
Using Jacobi's method find the complete integral of the equation
2u₁xz + 3u₂y² + u₂²u₃ = 0
1
Expert's answer
2019-08-29T10:49:08-0400

Using Jacobi's method find the complete integral of the equation  


2u1xz+3u2z2+u22u3=02u_1xz+3u_2z^2+u_2^2u_3=0

This equation is already in the form of 


F(x,y,z,u1,u2,u3)=0F(x,y,z,u_1,u_2,u_3)=0

 Its auxiliary equations are 


dxFu1=dyFu2=dzFu3=du1Fx=du2Fy=du3Fz{dx \over F_{u_1} }={dy \over F_{u_2} }={dz \over F_{u_3} }={du_1 \over -F_x }={du_2 \over -F_y }={du_3 \over -F_z }

dx2xz=dy3z2+2u2u3=dzu22=du12u1z=du20=du32u1x6u2z{dx \over 2xz }={dy \over 3z^2+2u_2u_3 }={dz \over u_2^2 }={du_1 \over -2u_1z }={du_2 \over 0 }={du_3 \over -2u_1x-6u_2z }

From the first and fourth ratios, we get


dx2xz=du12u1z=>dxx=du1u1=>lnx+lnu1=lna{dx \over 2xz }={du_1 \over -2u_1z }=>-{dx \over x }={du_1 \over u_1 }=>\ln{x}+\ln{u_1}=\ln{a}u1x=a,a=constu_1x=a, a=const

From the fifth ratio, we get


du20=>du2=0=>u2=b,b=const{du_2 \over 0 }=>du_2=0=>u_2=b, b=const

With the above forms of u1u_1 and u2,u_2, we obtain from the given equation 


2az+3bz2+b2u3=02az+3bz^2+b^2u_3=0u3=2az+3bz2b2u_3=-{2az+3bz^2 \over b^2}

The equation


du=u1dx+u2dy+u3dzdu=u_1dx+u_2dy+u_3dz

obtained for these values of u1,u2,u3,u_1,u_2,u_3, is integrable. 


du=(ax)dx+bdy+(2az+3bz2b2)dzdu=({a \over x})dx+bdy+(-{2az+3bz^2 \over b^2})dz

On integrating the above equation, we arrive at the following solution of the given equation 


u=alnx+byaz2b2z3b+c,u=a\ln x+by-{az^2 \over b^2}-{z^3 \over b}+c,a=const,b=const,c=consta=const, b=const, c=const

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