4.22.35 \(4 y(x)^2 y'(x)^3-2 x y'(x)+y(x)=0\)

ODE
\[ 4 y(x)^2 y'(x)^3-2 x y'(x)+y(x)=0 \] ODE Classification

[[_1st_order, _with_linear_symmetries]]

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

Mathematica ✓
cpu = 14.3887 (sec), leaf count = 2323

\[\left \{\text {Solve}\left [c_1=\int _1^x-\frac {27\ 3^{2/3} \left (\sqrt {81 y(x)^{10}-24 K[1]^3 y(x)^6}-9 y(x)^5\right )^{2/3} y(x)^6-8\ 3^{2/3} K[1]^3 \left (\sqrt {81 y(x)^{10}-24 K[1]^3 y(x)^6}-9 y(x)^5\right )^{2/3} y(x)^2+9 \sqrt [6]{3} \sqrt {y(x)^6 \left (27 y(x)^4-8 K[1]^3\right )} \left (\sqrt {81 y(x)^{10}-24 K[1]^3 y(x)^6}-9 y(x)^5\right )^{2/3} y(x)+4\ 3^{5/6} K[1]^2 \sqrt {27 y(x)^{10}-8 K[1]^3 y(x)^6} \sqrt [3]{\sqrt {81 y(x)^{10}-24 K[1]^3 y(x)^6}-9 y(x)^5}}{6 K[1]^2 y(x)^4 \left (8 K[1]^3-27 y(x)^4\right )}dK[1]+\int _1^{y(x)}\frac {324 x K[2]^8-96 x^4 K[2]^4+16\ 3^{2/3} x^3 \left (\sqrt {81 K[2]^{10}-24 x^3 K[2]^6}-9 K[2]^5\right )^{2/3} K[2]^2-18 \sqrt [6]{3} \left (3 \sqrt {3} K[2]^5+\sqrt {K[2]^6 \left (27 K[2]^4-8 x^3\right )}\right ) \left (\sqrt {81 K[2]^{10}-24 x^3 K[2]^6}-9 K[2]^5\right )^{2/3} K[2]+9 \left (8 x^4 K[2]^5-27 x K[2]^9\right ) \int _1^x\frac {8 K[2]^8 \left (27\ 3^{5/6} K[2]^8+4\ 3^{5/6} K[1]^3 K[2]^4-9 \sqrt [3]{3} \sqrt {K[2]^6 \left (27 K[2]^4-8 K[1]^3\right )} K[2]^3+\sqrt [6]{3} K[1] \left (27 K[2]^5-\sqrt {81 K[2]^{10}-24 K[1]^3 K[2]^6}\right ) \sqrt [3]{\sqrt {81 K[2]^{10}-24 K[1]^3 K[2]^6}-9 K[2]^5}\right )}{\left (K[2]^6 \left (27 K[2]^4-8 K[1]^3\right )\right )^{3/2} \left (\sqrt {81 K[2]^{10}-24 K[1]^3 K[2]^6}-9 K[2]^5\right )^{2/3}}dK[1]-8\ 3^{5/6} x^2 \sqrt {27 K[2]^{10}-8 x^3 K[2]^6} \sqrt [3]{\sqrt {81 K[2]^{10}-24 x^3 K[2]^6}-9 K[2]^5}}{9 K[2]^5 \left (27 x K[2]^4-8 x^4\right )}dK[2],y(x)\right ],\text {Solve}\left [c_1=\int _1^x\frac {\sqrt [6]{3} \sqrt [3]{\sqrt {81 y(x)^{10}-24 K[3]^3 y(x)^6}-9 y(x)^5} \left (27 \left (-3 i+\sqrt {3}\right ) \sqrt [3]{\sqrt {81 y(x)^{10}-24 K[3]^3 y(x)^6}-9 y(x)^5} y(x)^6-8 \left (-3 i+\sqrt {3}\right ) K[3]^3 \sqrt [3]{\sqrt {81 y(x)^{10}-24 K[3]^3 y(x)^6}-9 y(x)^5} y(x)^2+9 \left (1-i \sqrt {3}\right ) \sqrt {y(x)^6 \left (27 y(x)^4-8 K[3]^3\right )} \sqrt [3]{\sqrt {81 y(x)^{10}-24 K[3]^3 y(x)^6}-9 y(x)^5} y(x)+4 \sqrt [6]{3} \left (3 i+\sqrt {3}\right ) K[3]^2 \sqrt {y(x)^6 \left (27 y(x)^4-8 K[3]^3\right )}\right )}{96 K[3]^5 y(x)^4-324 K[3]^2 y(x)^8}dK[3]+\int _1^{y(x)}\frac {324 x K[4]^8-96 x^4 K[4]^4-8 \sqrt [6]{3} \left (-3 i+\sqrt {3}\right ) x^3 \left (\sqrt {81 K[4]^{10}-24 x^3 K[4]^6}-9 K[4]^5\right )^{2/3} K[4]^2+9 \sqrt [6]{3} \left (3 \left (-3 i+\sqrt {3}\right ) K[4]^5+\left (1-i \sqrt {3}\right ) \sqrt {K[4]^6 \left (27 K[4]^4-8 x^3\right )}\right ) \left (\sqrt {81 K[4]^{10}-24 x^3 K[4]^6}-9 K[4]^5\right )^{2/3} K[4]+9 \left (8 x^4 K[4]^5-27 x K[4]^9\right ) \int _1^x-\frac {4 K[4]^8 \left (4 \sqrt [3]{3} \left (3 i+\sqrt {3}\right ) K[3]^3 K[4]^4+9 \sqrt [3]{3} \left (3 \left (3 i+\sqrt {3}\right ) K[4]^5-i \left (-i+\sqrt {3}\right ) \sqrt {K[4]^6 \left (27 K[4]^4-8 K[3]^3\right )}\right ) K[4]^3+\sqrt [6]{3} K[3] \left (27 \left (1-i \sqrt {3}\right ) K[4]^5-\left (-3 i+\sqrt {3}\right ) \sqrt {K[4]^6 \left (27 K[4]^4-8 K[3]^3\right )}\right ) \sqrt [3]{\sqrt {81 K[4]^{10}-24 K[3]^3 K[4]^6}-9 K[4]^5}\right )}{\left (K[4]^6 \left (27 K[4]^4-8 K[3]^3\right )\right )^{3/2} \left (\sqrt {81 K[4]^{10}-24 K[3]^3 K[4]^6}-9 K[4]^5\right )^{2/3}}dK[3]+4 \sqrt [3]{3} \left (3 i+\sqrt {3}\right ) x^2 \sqrt {K[4]^6 \left (27 K[4]^4-8 x^3\right )} \sqrt [3]{\sqrt {81 K[4]^{10}-24 x^3 K[4]^6}-9 K[4]^5}}{9 K[4]^5 \left (27 x K[4]^4-8 x^4\right )}dK[4],y(x)\right ],\text {Solve}\left [c_1=\int _1^x\frac {\sqrt [6]{3} \sqrt [3]{\sqrt {81 y(x)^{10}-24 K[5]^3 y(x)^6}-9 y(x)^5} \left (27 \left (3 i+\sqrt {3}\right ) \sqrt [3]{\sqrt {81 y(x)^{10}-24 K[5]^3 y(x)^6}-9 y(x)^5} y(x)^6-8 \left (3 i+\sqrt {3}\right ) K[5]^3 \sqrt [3]{\sqrt {81 y(x)^{10}-24 K[5]^3 y(x)^6}-9 y(x)^5} y(x)^2+9 \left (1+i \sqrt {3}\right ) \sqrt {y(x)^6 \left (27 y(x)^4-8 K[5]^3\right )} \sqrt [3]{\sqrt {81 y(x)^{10}-24 K[5]^3 y(x)^6}-9 y(x)^5} y(x)+4 \sqrt [6]{3} \left (-3 i+\sqrt {3}\right ) K[5]^2 \sqrt {y(x)^6 \left (27 y(x)^4-8 K[5]^3\right )}\right )}{96 K[5]^5 y(x)^4-324 K[5]^2 y(x)^8}dK[5]+\int _1^{y(x)}\frac {324 x K[6]^8-96 x^4 K[6]^4-8 \sqrt [6]{3} \left (3 i+\sqrt {3}\right ) x^3 \left (\sqrt {81 K[6]^{10}-24 x^3 K[6]^6}-9 K[6]^5\right )^{2/3} K[6]^2+9 \sqrt [6]{3} \left (3 \left (3 i+\sqrt {3}\right ) K[6]^5+\left (1+i \sqrt {3}\right ) \sqrt {K[6]^6 \left (27 K[6]^4-8 x^3\right )}\right ) \left (\sqrt {81 K[6]^{10}-24 x^3 K[6]^6}-9 K[6]^5\right )^{2/3} K[6]+9 \left (8 x^4 K[6]^5-27 x K[6]^9\right ) \int _1^x\frac {4 K[6]^8 \left (-4 \sqrt [3]{3} \left (-3 i+\sqrt {3}\right ) K[5]^3 K[6]^4+9 \sqrt [3]{3} \left (\left (1-i \sqrt {3}\right ) \sqrt {K[6]^6 \left (27 K[6]^4-8 K[5]^3\right )}-3 \left (-3 i+\sqrt {3}\right ) K[6]^5\right ) K[6]^3+\sqrt [6]{3} K[5] \left (\left (3 i+\sqrt {3}\right ) \sqrt {K[6]^6 \left (27 K[6]^4-8 K[5]^3\right )}-27 i \left (-i+\sqrt {3}\right ) K[6]^5\right ) \sqrt [3]{\sqrt {81 K[6]^{10}-24 K[5]^3 K[6]^6}-9 K[6]^5}\right )}{\left (K[6]^6 \left (27 K[6]^4-8 K[5]^3\right )\right )^{3/2} \left (\sqrt {81 K[6]^{10}-24 K[5]^3 K[6]^6}-9 K[6]^5\right )^{2/3}}dK[5]+4 \sqrt [3]{3} \left (-3 i+\sqrt {3}\right ) x^2 \sqrt {K[6]^6 \left (27 K[6]^4-8 x^3\right )} \sqrt [3]{\sqrt {81 K[6]^{10}-24 x^3 K[6]^6}-9 K[6]^5}}{9 K[6]^5 \left (27 x K[6]^4-8 x^4\right )}dK[6],y(x)\right ]\right \}\]

Maple ✓
cpu = 0.819 (sec), leaf count = 91

\[\left [y \left (x \right ) = -\frac {2^{\frac {3}{4}} 3^{\frac {1}{4}} x^{\frac {3}{4}}}{3}, y \left (x \right ) = \frac {2^{\frac {3}{4}} 3^{\frac {1}{4}} x^{\frac {3}{4}}}{3}, y \left (x \right ) = -\frac {i 2^{\frac {3}{4}} 3^{\frac {1}{4}} x^{\frac {3}{4}}}{3}, y \left (x \right ) = \frac {i 2^{\frac {3}{4}} 3^{\frac {1}{4}} x^{\frac {3}{4}}}{3}, y \left (x \right ) = \sqrt {-4 \textit {\_C1}^{3}+2 x \textit {\_C1}}, y \left (x \right ) = -\sqrt {-4 \textit {\_C1}^{3}+2 x \textit {\_C1}}\right ]\] Mathematica raw input

DSolve[y[x] - 2*x*y'[x] + 4*y[x]^2*y'[x]^3 == 0,y[x],x]

Mathematica raw output

{Solve[C[1] == Inactive[Integrate][-1/6*(4*3^(5/6)*K[1]^2*Sqrt[-8*K[1]^3*y[x]^6 
+ 27*y[x]^10]*(-9*y[x]^5 + Sqrt[-24*K[1]^3*y[x]^6 + 81*y[x]^10])^(1/3) - 8*3^(2/
3)*K[1]^3*y[x]^2*(-9*y[x]^5 + Sqrt[-24*K[1]^3*y[x]^6 + 81*y[x]^10])^(2/3) + 27*3
^(2/3)*y[x]^6*(-9*y[x]^5 + Sqrt[-24*K[1]^3*y[x]^6 + 81*y[x]^10])^(2/3) + 9*3^(1/
6)*y[x]*Sqrt[y[x]^6*(-8*K[1]^3 + 27*y[x]^4)]*(-9*y[x]^5 + Sqrt[-24*K[1]^3*y[x]^6
 + 81*y[x]^10])^(2/3))/(K[1]^2*y[x]^4*(8*K[1]^3 - 27*y[x]^4)), {K[1], 1, x}] + I
nactive[Integrate][(-96*x^4*K[2]^4 + 324*x*K[2]^8 - 8*3^(5/6)*x^2*Sqrt[-8*x^3*K[
2]^6 + 27*K[2]^10]*(-9*K[2]^5 + Sqrt[-24*x^3*K[2]^6 + 81*K[2]^10])^(1/3) + 16*3^
(2/3)*x^3*K[2]^2*(-9*K[2]^5 + Sqrt[-24*x^3*K[2]^6 + 81*K[2]^10])^(2/3) - 18*3^(1
/6)*K[2]*(3*Sqrt[3]*K[2]^5 + Sqrt[K[2]^6*(-8*x^3 + 27*K[2]^4)])*(-9*K[2]^5 + Sqr
t[-24*x^3*K[2]^6 + 81*K[2]^10])^(2/3) + 9*(8*x^4*K[2]^5 - 27*x*K[2]^9)*Inactive[
Integrate][(8*K[2]^8*(4*3^(5/6)*K[1]^3*K[2]^4 + 27*3^(5/6)*K[2]^8 - 9*3^(1/3)*K[
2]^3*Sqrt[K[2]^6*(-8*K[1]^3 + 27*K[2]^4)] + 3^(1/6)*K[1]*(27*K[2]^5 - Sqrt[-24*K
[1]^3*K[2]^6 + 81*K[2]^10])*(-9*K[2]^5 + Sqrt[-24*K[1]^3*K[2]^6 + 81*K[2]^10])^(
1/3)))/((K[2]^6*(-8*K[1]^3 + 27*K[2]^4))^(3/2)*(-9*K[2]^5 + Sqrt[-24*K[1]^3*K[2]
^6 + 81*K[2]^10])^(2/3)), {K[1], 1, x}])/(9*K[2]^5*(-8*x^4 + 27*x*K[2]^4)), {K[2
], 1, y[x]}], y[x]], Solve[C[1] == Inactive[Integrate][(3^(1/6)*(-9*y[x]^5 + Sqr
t[-24*K[3]^3*y[x]^6 + 81*y[x]^10])^(1/3)*(4*3^(1/6)*(3*I + Sqrt[3])*K[3]^2*Sqrt[
y[x]^6*(-8*K[3]^3 + 27*y[x]^4)] - 8*(-3*I + Sqrt[3])*K[3]^3*y[x]^2*(-9*y[x]^5 + 
Sqrt[-24*K[3]^3*y[x]^6 + 81*y[x]^10])^(1/3) + 27*(-3*I + Sqrt[3])*y[x]^6*(-9*y[x
]^5 + Sqrt[-24*K[3]^3*y[x]^6 + 81*y[x]^10])^(1/3) + 9*(1 - I*Sqrt[3])*y[x]*Sqrt[
y[x]^6*(-8*K[3]^3 + 27*y[x]^4)]*(-9*y[x]^5 + Sqrt[-24*K[3]^3*y[x]^6 + 81*y[x]^10
])^(1/3)))/(96*K[3]^5*y[x]^4 - 324*K[3]^2*y[x]^8), {K[3], 1, x}] + Inactive[Inte
grate][(-96*x^4*K[4]^4 + 324*x*K[4]^8 + 4*3^(1/3)*(3*I + Sqrt[3])*x^2*Sqrt[K[4]^
6*(-8*x^3 + 27*K[4]^4)]*(-9*K[4]^5 + Sqrt[-24*x^3*K[4]^6 + 81*K[4]^10])^(1/3) - 
8*3^(1/6)*(-3*I + Sqrt[3])*x^3*K[4]^2*(-9*K[4]^5 + Sqrt[-24*x^3*K[4]^6 + 81*K[4]
^10])^(2/3) + 9*3^(1/6)*K[4]*(3*(-3*I + Sqrt[3])*K[4]^5 + (1 - I*Sqrt[3])*Sqrt[K
[4]^6*(-8*x^3 + 27*K[4]^4)])*(-9*K[4]^5 + Sqrt[-24*x^3*K[4]^6 + 81*K[4]^10])^(2/
3) + 9*(8*x^4*K[4]^5 - 27*x*K[4]^9)*Inactive[Integrate][(-4*K[4]^8*(4*3^(1/3)*(3
*I + Sqrt[3])*K[3]^3*K[4]^4 + 9*3^(1/3)*K[4]^3*(3*(3*I + Sqrt[3])*K[4]^5 - I*(-I
 + Sqrt[3])*Sqrt[K[4]^6*(-8*K[3]^3 + 27*K[4]^4)]) + 3^(1/6)*K[3]*(27*(1 - I*Sqrt
[3])*K[4]^5 - (-3*I + Sqrt[3])*Sqrt[K[4]^6*(-8*K[3]^3 + 27*K[4]^4)])*(-9*K[4]^5 
+ Sqrt[-24*K[3]^3*K[4]^6 + 81*K[4]^10])^(1/3)))/((K[4]^6*(-8*K[3]^3 + 27*K[4]^4)
)^(3/2)*(-9*K[4]^5 + Sqrt[-24*K[3]^3*K[4]^6 + 81*K[4]^10])^(2/3)), {K[3], 1, x}]
)/(9*K[4]^5*(-8*x^4 + 27*x*K[4]^4)), {K[4], 1, y[x]}], y[x]], Solve[C[1] == Inac
tive[Integrate][(3^(1/6)*(-9*y[x]^5 + Sqrt[-24*K[5]^3*y[x]^6 + 81*y[x]^10])^(1/3
)*(4*3^(1/6)*(-3*I + Sqrt[3])*K[5]^2*Sqrt[y[x]^6*(-8*K[5]^3 + 27*y[x]^4)] - 8*(3
*I + Sqrt[3])*K[5]^3*y[x]^2*(-9*y[x]^5 + Sqrt[-24*K[5]^3*y[x]^6 + 81*y[x]^10])^(
1/3) + 27*(3*I + Sqrt[3])*y[x]^6*(-9*y[x]^5 + Sqrt[-24*K[5]^3*y[x]^6 + 81*y[x]^1
0])^(1/3) + 9*(1 + I*Sqrt[3])*y[x]*Sqrt[y[x]^6*(-8*K[5]^3 + 27*y[x]^4)]*(-9*y[x]
^5 + Sqrt[-24*K[5]^3*y[x]^6 + 81*y[x]^10])^(1/3)))/(96*K[5]^5*y[x]^4 - 324*K[5]^
2*y[x]^8), {K[5], 1, x}] + Inactive[Integrate][(-96*x^4*K[6]^4 + 324*x*K[6]^8 + 
4*3^(1/3)*(-3*I + Sqrt[3])*x^2*Sqrt[K[6]^6*(-8*x^3 + 27*K[6]^4)]*(-9*K[6]^5 + Sq
rt[-24*x^3*K[6]^6 + 81*K[6]^10])^(1/3) - 8*3^(1/6)*(3*I + Sqrt[3])*x^3*K[6]^2*(-
9*K[6]^5 + Sqrt[-24*x^3*K[6]^6 + 81*K[6]^10])^(2/3) + 9*3^(1/6)*K[6]*(3*(3*I + S
qrt[3])*K[6]^5 + (1 + I*Sqrt[3])*Sqrt[K[6]^6*(-8*x^3 + 27*K[6]^4)])*(-9*K[6]^5 +
 Sqrt[-24*x^3*K[6]^6 + 81*K[6]^10])^(2/3) + 9*(8*x^4*K[6]^5 - 27*x*K[6]^9)*Inact
ive[Integrate][(4*K[6]^8*(-4*3^(1/3)*(-3*I + Sqrt[3])*K[5]^3*K[6]^4 + 9*3^(1/3)*
K[6]^3*(-3*(-3*I + Sqrt[3])*K[6]^5 + (1 - I*Sqrt[3])*Sqrt[K[6]^6*(-8*K[5]^3 + 27
*K[6]^4)]) + 3^(1/6)*K[5]*((-27*I)*(-I + Sqrt[3])*K[6]^5 + (3*I + Sqrt[3])*Sqrt[
K[6]^6*(-8*K[5]^3 + 27*K[6]^4)])*(-9*K[6]^5 + Sqrt[-24*K[5]^3*K[6]^6 + 81*K[6]^1
0])^(1/3)))/((K[6]^6*(-8*K[5]^3 + 27*K[6]^4))^(3/2)*(-9*K[6]^5 + Sqrt[-24*K[5]^3
*K[6]^6 + 81*K[6]^10])^(2/3)), {K[5], 1, x}])/(9*K[6]^5*(-8*x^4 + 27*x*K[6]^4)),
 {K[6], 1, y[x]}], y[x]]}

Maple raw input

dsolve(4*y(x)^2*diff(y(x),x)^3-2*x*diff(y(x),x)+y(x) = 0, y(x))

Maple raw output

[y(x) = -1/3*2^(3/4)*3^(1/4)*x^(3/4), y(x) = 1/3*2^(3/4)*3^(1/4)*x^(3/4), y(x) =
 -1/3*I*2^(3/4)*3^(1/4)*x^(3/4), y(x) = 1/3*I*2^(3/4)*3^(1/4)*x^(3/4), y(x) = (-
4*_C1^3+2*_C1*x)^(1/2), y(x) = -(-4*_C1^3+2*_C1*x)^(1/2)]