Solving Cauchy Problem: General Solution of xy3zx+x2z2zy=y3z

[senan]

Homework Statement

getting gen sol of

xy³z_x+x²z²z_y=y³z

solve cauchy problem
x=y=t, z=1/t

The Attempt at a Solution

i got gen sol F(C1,C2)=0 as

C1=x/z, C2=y⁴-x²z²

i inserted t for x and y and 1/t for z and ended up with

C1-2=1/(C2²)

I'm unsure what to do from here i tried to get equation in terms of z by inserting eqns for C1 and C2 but it becomes very messy

 

[jackmell]

Try to learn to not let messy intimidate you. Just muscle through it if you have to but you don't have to in this case. Just leave it in it's implicit form:

$$F(C1,C2)=F(x/y,y^4-x^2 z^2)=0$$

and often you won't be able to solve explicitly for z(x,y) anyway.

Here's a back-substitution check of $C1+C2^2=0$ in Mathematica. Note how $y^3 z$ is obtained after evaluating the left side as expected.

try and follow it and change the expression for F(C1,C2) if you wish:

In[40]:=
c1 = x/z[x, y]; 
c2 = y^4 - x^2*z[x, y]^2; 
myexp = c1 + c2^2 == 0; 
myzx = First[D[z[x, y], x] /. 
    Solve[D[myexp, x], D[z[x, y], x]]]
myzy = First[D[z[x, y], y] /. 
    Solve[D[myexp, y], D[z[x, y], y]]]
Simplify[x*y^3*myzx + x^2*z[x, y]^2*myzy]

Out[43]=
-((z[x, y]*(1 - 4*x*y^4*z[x, y]^3 + 
     4*x^3*z[x, y]^5))/
   (x*(-1 - 4*x*y^4*z[x, y]^3 + 
     4*x^3*z[x, y]^5)))

Out[44]=
-((8*y^3*z[x, y]^2*(y^4 - x^2*z[x, y]^2))/
   (x*(-1 - 4*x*y^4*z[x, y]^3 + 
     4*x^3*z[x, y]^5)))

Out[45]=
y^3*z[x, y]