4.19.32 $x^3{y}^{\prime }(x{)}^2+x{y}^{\prime }(x)-y(x)=0$

ODE
$$x^3{y}^{\prime }(x{)}^2+x{y}^{\prime }(x)-y(x)=0$$ ODE Classification

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

Book solution method
No Missing Variables ODE, Solve for $y$

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

$$\{\{y(x)\to -\frac{\sqrt{3}\sqrt{-\frac{-\frac{c_1^4{x}^6}{\sqrt[3]{93312x^5{c}_1^{10}+2160x^7{c}_1^8-x^9{c}_1^6+48\sqrt{3}\sqrt{-x^{10}{c}_1^{14}(x^2-108c_1^2){}^3}}}-\frac{864c_1^6{x}^4}{\sqrt[3]{93312x^5{c}_1^{10}+2160x^7{c}_1^8-x^9{c}_1^6+48\sqrt{3}\sqrt{-x^{10}{c}_1^{14}(x^2-108c_1^2){}^3}}}+c_1^2{x}^3-48c_1^{4}x-\sqrt[3]{93312x^5{c}_1^{10}+2160x^7{c}_1^8-x^9{c}_1^6+48\sqrt{3}\sqrt{-x^{10}{c}_1^{14}(x^2-108c_1^2){}^3}}}{x^3{c}_1^4}}x+\sqrt{3}\sqrt{\frac{96\sqrt{3}(x^2+4c_1^2)}{x^3{c}_1^2\sqrt{-\frac{-\frac{c_1^4{x}^6}{\sqrt[3]{93312x^5{c}_1^{10}+2160x^7{c}_1^8-x^9{c}_1^6+48\sqrt{3}\sqrt{-x^{10}{c}_1^{14}(x^2-108c_1^2){}^3}}}-\frac{864c_1^6{x}^4}{\sqrt[3]{93312x^5{c}_1^{10}+2160x^7{c}_1^8-x^9{c}_1^6+48\sqrt{3}\sqrt{-x^{10}{c}_1^{14}(x^2-108c_1^2){}^3}}}+c_1^2{x}^3-48c_1^{4}x-\sqrt[3]{93312x^5{c}_1^{10}+2160x^7{c}_1^8-x^9{c}_1^6+48\sqrt{3}\sqrt{-x^{10}{c}_1^{14}(x^2-108c_1^2){}^3}}}{x^3{c}_1^4}}}-\frac{\sqrt[3]{93312x^5{c}_1^{10}+2160x^7{c}_1^8-x^9{c}_1^6+48\sqrt{3}\sqrt{-x^{10}{c}_1^{14}(x^2-108c_1^2){}^3}}}{x^3{c}_1^4}-\frac{x(x^2+864c_1^2)}{\sqrt[3]{93312x^5{c}_1^{10}+2160x^7{c}_1^8-x^9{c}_1^6+48\sqrt{3}\sqrt{-x^{10}{c}_1^{14}(x^2-108c_1^2){}^3}}}+\frac{96}{x^2}-\frac{2}{c_1^2}}x-36}{24x}\},\{y(x)\to \frac{-\sqrt{3}\sqrt{-\frac{-\frac{c_1^4{x}^6}{\sqrt[3]{93312x^5{c}_1^{10}+2160x^7{c}_1^8-x^9{c}_1^6+48\sqrt{3}\sqrt{-x^{10}{c}_1^{14}(x^2-108c_1^2){}^3}}}-\frac{864c_1^6{x}^4}{\sqrt[3]{93312x^5{c}_1^{10}+2160x^7{c}_1^8-x^9{c}_1^6+48\sqrt{3}\sqrt{-x^{10}{c}_1^{14}(x^2-108c_1^2){}^3}}}+c_1^2{x}^3-48c_1^{4}x-\sqrt[3]{93312x^5{c}_1^{10}+2160x^7{c}_1^8-x^9{c}_1^6+48\sqrt{3}\sqrt{-x^{10}{c}_1^{14}(x^2-108c_1^2){}^3}}}{x^3{c}_1^4}}x+\sqrt{3}\sqrt{\frac{96\sqrt{3}(x^2+4c_1^2)}{x^3{c}_1^2\sqrt{-\frac{-\frac{c_1^4{x}^6}{\sqrt[3]{93312x^5{c}_1^{10}+2160x^7{c}_1^8-x^9{c}_1^6+48\sqrt{3}\sqrt{-x^{10}{c}_1^{14}(x^2-108c_1^2){}^3}}}-\frac{864c_1^6{x}^4}{\sqrt[3]{93312x^5{c}_1^{10}+2160x^7{c}_1^8-x^9{c}_1^6+48\sqrt{3}\sqrt{-x^{10}{c}_1^{14}(x^2-108c_1^2){}^3}}}+c_1^2{x}^3-48c_1^{4}x-\sqrt[3]{93312x^5{c}_1^{10}+2160x^7{c}_1^8-x^9{c}_1^6+48\sqrt{3}\sqrt{-x^{10}{c}_1^{14}(x^2-108c_1^2){}^3}}}{x^3{c}_1^4}}}-\frac{\sqrt[3]{93312x^5{c}_1^{10}+2160x^7{c}_1^8-x^9{c}_1^6+48\sqrt{3}\sqrt{-x^{10}{c}_1^{14}(x^2-108c_1^2){}^3}}}{x^3{c}_1^4}-\frac{x(x^2+864c_1^2)}{\sqrt[3]{93312x^5{c}_1^{10}+2160x^7{c}_1^8-x^9{c}_1^6+48\sqrt{3}\sqrt{-x^{10}{c}_1^{14}(x^2-108c_1^2){}^3}}}+\frac{96}{x^2}-\frac{2}{c_1^2}}x+36}{24x}\},\{y(x)\to \frac{\sqrt{3}\sqrt{-\frac{-\frac{c_1^4{x}^6}{\sqrt[3]{93312x^5{c}_1^{10}+2160x^7{c}_1^8-x^9{c}_1^6+48\sqrt{3}\sqrt{-x^{10}{c}_1^{14}(x^2-108c_1^2){}^3}}}-\frac{864c_1^6{x}^4}{\sqrt[3]{93312x^5{c}_1^{10}+2160x^7{c}_1^8-x^9{c}_1^6+48\sqrt{3}\sqrt{-x^{10}{c}_1^{14}(x^2-108c_1^2){}^3}}}+c_1^2{x}^3-48c_1^{4}x-\sqrt[3]{93312x^5{c}_1^{10}+2160x^7{c}_1^8-x^9{c}_1^6+48\sqrt{3}\sqrt{-x^{10}{c}_1^{14}(x^2-108c_1^2){}^3}}}{x^3{c}_1^4}}x-\sqrt{3}\sqrt{-\frac{96\sqrt{3}(x^2+4c_1^2)}{x^3{c}_1^2\sqrt{-\frac{-\frac{c_1^4{x}^6}{\sqrt[3]{93312x^5{c}_1^{10}+2160x^7{c}_1^8-x^9{c}_1^6+48\sqrt{3}\sqrt{-x^{10}{c}_1^{14}(x^2-108c_1^2){}^3}}}-\frac{864c_1^6{x}^4}{\sqrt[3]{93312x^5{c}_1^{10}+2160x^7{c}_1^8-x^9{c}_1^6+48\sqrt{3}\sqrt{-x^{10}{c}_1^{14}(x^2-108c_1^2){}^3}}}+c_1^2{x}^3-48c_1^{4}x-\sqrt[3]{93312x^5{c}_1^{10}+2160x^7{c}_1^8-x^9{c}_1^6+48\sqrt{3}\sqrt{-x^{10}{c}_1^{14}(x^2-108c_1^2){}^3}}}{x^3{c}_1^4}}}-\frac{\sqrt[3]{93312x^5{c}_1^{10}+2160x^7{c}_1^8-x^9{c}_1^6+48\sqrt{3}\sqrt{-x^{10}{c}_1^{14}(x^2-108c_1^2){}^3}}}{x^3{c}_1^4}-\frac{x(x^2+864c_1^2)}{\sqrt[3]{93312x^5{c}_1^{10}+2160x^7{c}_1^8-x^9{c}_1^6+48\sqrt{3}\sqrt{-x^{10}{c}_1^{14}(x^2-108c_1^2){}^3}}}+\frac{96}{x^2}-\frac{2}{c_1^2}}x+36}{24x}\},\{y(x)\to \frac{\sqrt{3}\sqrt{-\frac{-\frac{c_1^4{x}^6}{\sqrt[3]{93312x^5{c}_1^{10}+2160x^7{c}_1^8-x^9{c}_1^6+48\sqrt{3}\sqrt{-x^{10}{c}_1^{14}(x^2-108c_1^2){}^3}}}-\frac{864c_1^6{x}^4}{\sqrt[3]{93312x^5{c}_1^{10}+2160x^7{c}_1^8-x^9{c}_1^6+48\sqrt{3}\sqrt{-x^{10}{c}_1^{14}(x^2-108c_1^2){}^3}}}+c_1^2{x}^3-48c_1^{4}x-\sqrt[3]{93312x^5{c}_1^{10}+2160x^7{c}_1^8-x^9{c}_1^6+48\sqrt{3}\sqrt{-x^{10}{c}_1^{14}(x^2-108c_1^2){}^3}}}{x^3{c}_1^4}}x+\sqrt{3}\sqrt{-\frac{96\sqrt{3}(x^2+4c_1^2)}{x^3{c}_1^2\sqrt{-\frac{-\frac{c_1^4{x}^6}{\sqrt[3]{93312x^5{c}_1^{10}+2160x^7{c}_1^8-x^9{c}_1^6+48\sqrt{3}\sqrt{-x^{10}{c}_1^{14}(x^2-108c_1^2){}^3}}}-\frac{864c_1^6{x}^4}{\sqrt[3]{93312x^5{c}_1^{10}+2160x^7{c}_1^8-x^9{c}_1^6+48\sqrt{3}\sqrt{-x^{10}{c}_1^{14}(x^2-108c_1^2){}^3}}}+c_1^2{x}^3-48c_1^{4}x-\sqrt[3]{93312x^5{c}_1^{10}+2160x^7{c}_1^8-x^9{c}_1^6+48\sqrt{3}\sqrt{-x^{10}{c}_1^{14}(x^2-108c_1^2){}^3}}}{x^3{c}_1^4}}}-\frac{\sqrt[3]{93312x^5{c}_1^{10}+2160x^7{c}_1^8-x^9{c}_1^6+48\sqrt{3}\sqrt{-x^{10}{c}_1^{14}(x^2-108c_1^2){}^3}}}{x^3{c}_1^4}-\frac{x(x^2+864c_1^2)}{\sqrt[3]{93312x^5{c}_1^{10}+2160x^7{c}_1^8-x^9{c}_1^6+48\sqrt{3}\sqrt{-x^{10}{c}_1^{14}(x^2-108c_1^2){}^3}}}+\frac{96}{x^2}-\frac{2}{c_1^2}}x+36}{24x}\}\}$$

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

$$\{\ln \left(x\right)-\mathit{\_}\mathit{C}\mathit{1}+\frac{1}{4}\ln (-1+\sqrt{1+4xy\left(x\right)})-\frac{3}{4}\ln (\sqrt{1+4xy\left(x\right)}-3)+\frac{3}{4}\ln (\sqrt{1+4xy\left(x\right)}+3)-\frac{1}{4}\ln (1+\sqrt{1+4xy\left(x\right)})-\frac{3\ln (xy\left(x\right)-2)}{4}-\frac{\ln \left(xy\left(x\right)\right)}{4}=0,\ln \left(x\right)-\mathit{\_}\mathit{C}\mathit{1}-\frac{3\ln (xy\left(x\right)-2)}{4}-\frac{\ln \left(xy\left(x\right)\right)}{4}-\frac{1}{4}\ln (-1+\sqrt{1+4xy\left(x\right)})+\frac{3}{4}\ln (\sqrt{1+4xy\left(x\right)}-3)-\frac{3}{4}\ln (\sqrt{1+4xy\left(x\right)}+3)+\frac{1}{4}\ln (1+\sqrt{1+4xy\left(x\right)})=0\}$$ 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