4.19.32 \(x^3 y'(x)^2+x y'(x)-y(x)=0\)

ODE
\[ x^3 y'(x)^2+x y'(x)-y(x)=0 \] ODE Classification

[[_homogeneous, `class G`], _rational]

Book solution method
No Missing Variables ODE, Solve for \(y\)

Mathematica
cpu = 1.39819 (sec), leaf count = 2799

\[\left \{\left \{y(x)\to -\frac {\sqrt {3} \sqrt {-\frac {-\frac {c_1^4 x^6}{\sqrt [3]{93312 x^5 c_1^{10}+2160 x^7 c_1^8-x^9 c_1^6+48 \sqrt {3} \sqrt {-x^{10} c_1^{14} \left (x^2-108 c_1^2\right ){}^3}}}-\frac {864 c_1^6 x^4}{\sqrt [3]{93312 x^5 c_1^{10}+2160 x^7 c_1^8-x^9 c_1^6+48 \sqrt {3} \sqrt {-x^{10} c_1^{14} \left (x^2-108 c_1^2\right ){}^3}}}+c_1^2 x^3-48 c_1^4 x-\sqrt [3]{93312 x^5 c_1^{10}+2160 x^7 c_1^8-x^9 c_1^6+48 \sqrt {3} \sqrt {-x^{10} c_1^{14} \left (x^2-108 c_1^2\right ){}^3}}}{x^3 c_1^4}} x+\sqrt {3} \sqrt {\frac {96 \sqrt {3} \left (x^2+4 c_1^2\right )}{x^3 c_1^2 \sqrt {-\frac {-\frac {c_1^4 x^6}{\sqrt [3]{93312 x^5 c_1^{10}+2160 x^7 c_1^8-x^9 c_1^6+48 \sqrt {3} \sqrt {-x^{10} c_1^{14} \left (x^2-108 c_1^2\right ){}^3}}}-\frac {864 c_1^6 x^4}{\sqrt [3]{93312 x^5 c_1^{10}+2160 x^7 c_1^8-x^9 c_1^6+48 \sqrt {3} \sqrt {-x^{10} c_1^{14} \left (x^2-108 c_1^2\right ){}^3}}}+c_1^2 x^3-48 c_1^4 x-\sqrt [3]{93312 x^5 c_1^{10}+2160 x^7 c_1^8-x^9 c_1^6+48 \sqrt {3} \sqrt {-x^{10} c_1^{14} \left (x^2-108 c_1^2\right ){}^3}}}{x^3 c_1^4}}}-\frac {\sqrt [3]{93312 x^5 c_1^{10}+2160 x^7 c_1^8-x^9 c_1^6+48 \sqrt {3} \sqrt {-x^{10} c_1^{14} \left (x^2-108 c_1^2\right ){}^3}}}{x^3 c_1^4}-\frac {x \left (x^2+864 c_1^2\right )}{\sqrt [3]{93312 x^5 c_1^{10}+2160 x^7 c_1^8-x^9 c_1^6+48 \sqrt {3} \sqrt {-x^{10} c_1^{14} \left (x^2-108 c_1^2\right ){}^3}}}+\frac {96}{x^2}-\frac {2}{c_1^2}} x-36}{24 x}\right \},\left \{y(x)\to \frac {-\sqrt {3} \sqrt {-\frac {-\frac {c_1^4 x^6}{\sqrt [3]{93312 x^5 c_1^{10}+2160 x^7 c_1^8-x^9 c_1^6+48 \sqrt {3} \sqrt {-x^{10} c_1^{14} \left (x^2-108 c_1^2\right ){}^3}}}-\frac {864 c_1^6 x^4}{\sqrt [3]{93312 x^5 c_1^{10}+2160 x^7 c_1^8-x^9 c_1^6+48 \sqrt {3} \sqrt {-x^{10} c_1^{14} \left (x^2-108 c_1^2\right ){}^3}}}+c_1^2 x^3-48 c_1^4 x-\sqrt [3]{93312 x^5 c_1^{10}+2160 x^7 c_1^8-x^9 c_1^6+48 \sqrt {3} \sqrt {-x^{10} c_1^{14} \left (x^2-108 c_1^2\right ){}^3}}}{x^3 c_1^4}} x+\sqrt {3} \sqrt {\frac {96 \sqrt {3} \left (x^2+4 c_1^2\right )}{x^3 c_1^2 \sqrt {-\frac {-\frac {c_1^4 x^6}{\sqrt [3]{93312 x^5 c_1^{10}+2160 x^7 c_1^8-x^9 c_1^6+48 \sqrt {3} \sqrt {-x^{10} c_1^{14} \left (x^2-108 c_1^2\right ){}^3}}}-\frac {864 c_1^6 x^4}{\sqrt [3]{93312 x^5 c_1^{10}+2160 x^7 c_1^8-x^9 c_1^6+48 \sqrt {3} \sqrt {-x^{10} c_1^{14} \left (x^2-108 c_1^2\right ){}^3}}}+c_1^2 x^3-48 c_1^4 x-\sqrt [3]{93312 x^5 c_1^{10}+2160 x^7 c_1^8-x^9 c_1^6+48 \sqrt {3} \sqrt {-x^{10} c_1^{14} \left (x^2-108 c_1^2\right ){}^3}}}{x^3 c_1^4}}}-\frac {\sqrt [3]{93312 x^5 c_1^{10}+2160 x^7 c_1^8-x^9 c_1^6+48 \sqrt {3} \sqrt {-x^{10} c_1^{14} \left (x^2-108 c_1^2\right ){}^3}}}{x^3 c_1^4}-\frac {x \left (x^2+864 c_1^2\right )}{\sqrt [3]{93312 x^5 c_1^{10}+2160 x^7 c_1^8-x^9 c_1^6+48 \sqrt {3} \sqrt {-x^{10} c_1^{14} \left (x^2-108 c_1^2\right ){}^3}}}+\frac {96}{x^2}-\frac {2}{c_1^2}} x+36}{24 x}\right \},\left \{y(x)\to \frac {\sqrt {3} \sqrt {-\frac {-\frac {c_1^4 x^6}{\sqrt [3]{93312 x^5 c_1^{10}+2160 x^7 c_1^8-x^9 c_1^6+48 \sqrt {3} \sqrt {-x^{10} c_1^{14} \left (x^2-108 c_1^2\right ){}^3}}}-\frac {864 c_1^6 x^4}{\sqrt [3]{93312 x^5 c_1^{10}+2160 x^7 c_1^8-x^9 c_1^6+48 \sqrt {3} \sqrt {-x^{10} c_1^{14} \left (x^2-108 c_1^2\right ){}^3}}}+c_1^2 x^3-48 c_1^4 x-\sqrt [3]{93312 x^5 c_1^{10}+2160 x^7 c_1^8-x^9 c_1^6+48 \sqrt {3} \sqrt {-x^{10} c_1^{14} \left (x^2-108 c_1^2\right ){}^3}}}{x^3 c_1^4}} x-\sqrt {3} \sqrt {-\frac {96 \sqrt {3} \left (x^2+4 c_1^2\right )}{x^3 c_1^2 \sqrt {-\frac {-\frac {c_1^4 x^6}{\sqrt [3]{93312 x^5 c_1^{10}+2160 x^7 c_1^8-x^9 c_1^6+48 \sqrt {3} \sqrt {-x^{10} c_1^{14} \left (x^2-108 c_1^2\right ){}^3}}}-\frac {864 c_1^6 x^4}{\sqrt [3]{93312 x^5 c_1^{10}+2160 x^7 c_1^8-x^9 c_1^6+48 \sqrt {3} \sqrt {-x^{10} c_1^{14} \left (x^2-108 c_1^2\right ){}^3}}}+c_1^2 x^3-48 c_1^4 x-\sqrt [3]{93312 x^5 c_1^{10}+2160 x^7 c_1^8-x^9 c_1^6+48 \sqrt {3} \sqrt {-x^{10} c_1^{14} \left (x^2-108 c_1^2\right ){}^3}}}{x^3 c_1^4}}}-\frac {\sqrt [3]{93312 x^5 c_1^{10}+2160 x^7 c_1^8-x^9 c_1^6+48 \sqrt {3} \sqrt {-x^{10} c_1^{14} \left (x^2-108 c_1^2\right ){}^3}}}{x^3 c_1^4}-\frac {x \left (x^2+864 c_1^2\right )}{\sqrt [3]{93312 x^5 c_1^{10}+2160 x^7 c_1^8-x^9 c_1^6+48 \sqrt {3} \sqrt {-x^{10} c_1^{14} \left (x^2-108 c_1^2\right ){}^3}}}+\frac {96}{x^2}-\frac {2}{c_1^2}} x+36}{24 x}\right \},\left \{y(x)\to \frac {\sqrt {3} \sqrt {-\frac {-\frac {c_1^4 x^6}{\sqrt [3]{93312 x^5 c_1^{10}+2160 x^7 c_1^8-x^9 c_1^6+48 \sqrt {3} \sqrt {-x^{10} c_1^{14} \left (x^2-108 c_1^2\right ){}^3}}}-\frac {864 c_1^6 x^4}{\sqrt [3]{93312 x^5 c_1^{10}+2160 x^7 c_1^8-x^9 c_1^6+48 \sqrt {3} \sqrt {-x^{10} c_1^{14} \left (x^2-108 c_1^2\right ){}^3}}}+c_1^2 x^3-48 c_1^4 x-\sqrt [3]{93312 x^5 c_1^{10}+2160 x^7 c_1^8-x^9 c_1^6+48 \sqrt {3} \sqrt {-x^{10} c_1^{14} \left (x^2-108 c_1^2\right ){}^3}}}{x^3 c_1^4}} x+\sqrt {3} \sqrt {-\frac {96 \sqrt {3} \left (x^2+4 c_1^2\right )}{x^3 c_1^2 \sqrt {-\frac {-\frac {c_1^4 x^6}{\sqrt [3]{93312 x^5 c_1^{10}+2160 x^7 c_1^8-x^9 c_1^6+48 \sqrt {3} \sqrt {-x^{10} c_1^{14} \left (x^2-108 c_1^2\right ){}^3}}}-\frac {864 c_1^6 x^4}{\sqrt [3]{93312 x^5 c_1^{10}+2160 x^7 c_1^8-x^9 c_1^6+48 \sqrt {3} \sqrt {-x^{10} c_1^{14} \left (x^2-108 c_1^2\right ){}^3}}}+c_1^2 x^3-48 c_1^4 x-\sqrt [3]{93312 x^5 c_1^{10}+2160 x^7 c_1^8-x^9 c_1^6+48 \sqrt {3} \sqrt {-x^{10} c_1^{14} \left (x^2-108 c_1^2\right ){}^3}}}{x^3 c_1^4}}}-\frac {\sqrt [3]{93312 x^5 c_1^{10}+2160 x^7 c_1^8-x^9 c_1^6+48 \sqrt {3} \sqrt {-x^{10} c_1^{14} \left (x^2-108 c_1^2\right ){}^3}}}{x^3 c_1^4}-\frac {x \left (x^2+864 c_1^2\right )}{\sqrt [3]{93312 x^5 c_1^{10}+2160 x^7 c_1^8-x^9 c_1^6+48 \sqrt {3} \sqrt {-x^{10} c_1^{14} \left (x^2-108 c_1^2\right ){}^3}}}+\frac {96}{x^2}-\frac {2}{c_1^2}} x+36}{24 x}\right \}\right \}\]

Maple
cpu = 0.068 (sec), leaf count = 161

\[ \left \{ \ln \relax (x ) -{\it \_C1}+{\frac {1}{4}\ln \left (-1+\sqrt {1+4\,xy \relax (x ) } \right ) }-{\frac {3}{4}\ln \left (\sqrt {1+4\,xy \relax (x ) }-3 \right ) }+{\frac {3}{4}\ln \left (\sqrt {1+4\,xy \relax (x ) }+3 \right ) }-{\frac {1}{4}\ln \left (1+\sqrt {1+4\,xy \relax (x ) } \right ) }-{\frac {3\,\ln \left (xy \relax (x ) -2 \right ) }{4}}-{\frac {\ln \left (xy \relax (x ) \right ) }{4}}=0,\ln \relax (x ) -{\it \_C1}-{\frac {3\,\ln \left (xy \relax (x ) -2 \right ) }{4}}-{\frac {\ln \left (xy \relax (x ) \right ) }{4}}-{\frac {1}{4}\ln \left (-1+\sqrt {1+4\,xy \relax (x ) } \right ) }+{\frac {3}{4}\ln \left (\sqrt {1+4\,xy \relax (x ) }-3 \right ) }-{\frac {3}{4}\ln \left (\sqrt {1+4\,xy \relax (x ) }+3 \right ) }+{\frac {1}{4}\ln \left (1+\sqrt {1+4\,xy \relax (x ) } \right ) }=0 \right \} \] Mathematica raw input

DSolve[-y[x] + x*y'[x] + x^3*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> -(-36 + Sqrt[3]*x*Sqrt[-((x^3*C[1]^2 - 48*x*C[1]^4 - (x^6*C[1]^4)/(-(x
^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^1
4*(x^2 - 108*C[1]^2)^3)])^(1/3) - (864*x^4*C[1]^6)/(-(x^9*C[1]^6) + 2160*x^7*C[1
]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])
^(1/3) - (-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[
-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3))/(x^3*C[1]^4))] + Sqrt[3]*x*Sqrt[96
/x^2 - 2/C[1]^2 - (x*(x^2 + 864*C[1]^2))/(-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 9331
2*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3) - (
-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1
]^14*(x^2 - 108*C[1]^2)^3)])^(1/3)/(x^3*C[1]^4) + (96*Sqrt[3]*(x^2 + 4*C[1]^2))/
(x^3*C[1]^2*Sqrt[-((x^3*C[1]^2 - 48*x*C[1]^4 - (x^6*C[1]^4)/(-(x^9*C[1]^6) + 216
0*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1
]^2)^3)])^(1/3) - (864*x^4*C[1]^6)/(-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*
C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3) - (-(x^9*
C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(
x^2 - 108*C[1]^2)^3)])^(1/3))/(x^3*C[1]^4))])])/(24*x)}, {y[x] -> (36 - Sqrt[3]*
x*Sqrt[-((x^3*C[1]^2 - 48*x*C[1]^4 - (x^6*C[1]^4)/(-(x^9*C[1]^6) + 2160*x^7*C[1]
^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^
(1/3) - (864*x^4*C[1]^6)/(-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 
48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3) - (-(x^9*C[1]^6) + 
2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*
C[1]^2)^3)])^(1/3))/(x^3*C[1]^4))] + Sqrt[3]*x*Sqrt[96/x^2 - 2/C[1]^2 - (x*(x^2 
+ 864*C[1]^2))/(-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]
*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3) - (-(x^9*C[1]^6) + 2160*x^7*C
[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)
])^(1/3)/(x^3*C[1]^4) + (96*Sqrt[3]*(x^2 + 4*C[1]^2))/(x^3*C[1]^2*Sqrt[-((x^3*C[
1]^2 - 48*x*C[1]^4 - (x^6*C[1]^4)/(-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C
[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3) - (864*x^4
*C[1]^6)/(-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[
-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3) - (-(x^9*C[1]^6) + 2160*x^7*C[1]^8 
+ 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/
3))/(x^3*C[1]^4))])])/(24*x)}, {y[x] -> (36 + Sqrt[3]*x*Sqrt[-((x^3*C[1]^2 - 48*
x*C[1]^4 - (x^6*C[1]^4)/(-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 4
8*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3) - (864*x^4*C[1]^6)/(
-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1
]^14*(x^2 - 108*C[1]^2)^3)])^(1/3) - (-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^
5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3))/(x^3*C
[1]^4))] - Sqrt[3]*x*Sqrt[96/x^2 - 2/C[1]^2 - (x*(x^2 + 864*C[1]^2))/(-(x^9*C[1]
^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 
- 108*C[1]^2)^3)])^(1/3) - (-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 
+ 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3)/(x^3*C[1]^4) - (9
6*Sqrt[3]*(x^2 + 4*C[1]^2))/(x^3*C[1]^2*Sqrt[-((x^3*C[1]^2 - 48*x*C[1]^4 - (x^6*
C[1]^4)/(-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-
(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3) - (864*x^4*C[1]^6)/(-(x^9*C[1]^6) + 
2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*
C[1]^2)^3)])^(1/3) - (-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*S
qrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3))/(x^3*C[1]^4))])])/(24*
x)}, {y[x] -> (36 + Sqrt[3]*x*Sqrt[-((x^3*C[1]^2 - 48*x*C[1]^4 - (x^6*C[1]^4)/(-
(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]
^14*(x^2 - 108*C[1]^2)^3)])^(1/3) - (864*x^4*C[1]^6)/(-(x^9*C[1]^6) + 2160*x^7*C
[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)
])^(1/3) - (-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqr
t[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3))/(x^3*C[1]^4))] + Sqrt[3]*x*Sqrt[
96/x^2 - 2/C[1]^2 - (x*(x^2 + 864*C[1]^2))/(-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93
312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3) -
 (-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C
[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3)/(x^3*C[1]^4) - (96*Sqrt[3]*(x^2 + 4*C[1]^2)
)/(x^3*C[1]^2*Sqrt[-((x^3*C[1]^2 - 48*x*C[1]^4 - (x^6*C[1]^4)/(-(x^9*C[1]^6) + 2
160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C
[1]^2)^3)])^(1/3) - (864*x^4*C[1]^6)/(-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^
5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3) - (-(x^
9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14
*(x^2 - 108*C[1]^2)^3)])^(1/3))/(x^3*C[1]^4))])])/(24*x)}}

Maple raw input

dsolve(x^3*diff(y(x),x)^2+x*diff(y(x),x)-y(x) = 0, y(x),'implicit')

Maple raw output

ln(x)-_C1-3/4*ln(x*y(x)-2)-1/4*ln(x*y(x))-1/4*ln(-1+(1+4*x*y(x))^(1/2))+3/4*ln((
1+4*x*y(x))^(1/2)-3)-3/4*ln((1+4*x*y(x))^(1/2)+3)+1/4*ln(1+(1+4*x*y(x))^(1/2)) =
 0, ln(x)-_C1+1/4*ln(-1+(1+4*x*y(x))^(1/2))-3/4*ln((1+4*x*y(x))^(1/2)-3)+3/4*ln(
(1+4*x*y(x))^(1/2)+3)-1/4*ln(1+(1+4*x*y(x))^(1/2))-3/4*ln(x*y(x)-2)-1/4*ln(x*y(x
)) = 0