4.18.40 \(4 x y'(x)^2-3 y(x) y'(x)+3=0\)

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

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

Book solution method
Clairaut’s equation and related types, d’Alembert’s equation (also call Lagrange’s)

Mathematica
cpu = 1.15782 (sec), leaf count = 613

\[\left \{\left \{y(x)\to -\frac {1}{3} \sqrt {\frac {2}{3}} \sqrt {\left (\sinh \left (c_1\right )-\cosh \left (c_1\right )\right ) \sqrt {\left (\sinh \left (c_1\right )+\cosh \left (c_1\right )\right ) \left (\sinh \left (c_1\right )+\cosh \left (c_1\right )-18 x\right ){}^3}+\sinh \left (c_1\right )+\cosh \left (c_1\right )+54 x}\right \},\left \{y(x)\to \frac {1}{3} \sqrt {\frac {2}{3}} \sqrt {\left (\sinh \left (c_1\right )-\cosh \left (c_1\right )\right ) \sqrt {\left (\sinh \left (c_1\right )+\cosh \left (c_1\right )\right ) \left (\sinh \left (c_1\right )+\cosh \left (c_1\right )-18 x\right ){}^3}+\sinh \left (c_1\right )+\cosh \left (c_1\right )+54 x}\right \},\left \{y(x)\to -\frac {1}{3} \sqrt {\frac {2}{3}} \sqrt {\left (\cosh \left (c_1\right )-\sinh \left (c_1\right )\right ) \sqrt {\left (\sinh \left (c_1\right )+\cosh \left (c_1\right )\right ) \left (\sinh \left (c_1\right )+\cosh \left (c_1\right )-18 x\right ){}^3}+\sinh \left (c_1\right )+\cosh \left (c_1\right )+54 x}\right \},\left \{y(x)\to \frac {1}{3} \sqrt {\frac {2}{3}} \sqrt {\left (\cosh \left (c_1\right )-\sinh \left (c_1\right )\right ) \sqrt {\left (\sinh \left (c_1\right )+\cosh \left (c_1\right )\right ) \left (\sinh \left (c_1\right )+\cosh \left (c_1\right )-18 x\right ){}^3}+\sinh \left (c_1\right )+\cosh \left (c_1\right )+54 x}\right \},\left \{y(x)\to -\frac {1}{3} \sqrt {\frac {2}{3}} \sqrt {\cosh \left (c_1\right ) \sqrt {\left (\sinh \left (c_1\right )+\cosh \left (c_1\right )\right ) \left (\sinh \left (c_1\right )+\cosh \left (c_1\right )+18 x\right ){}^3}-\sinh \left (c_1\right ) \sqrt {\left (\sinh \left (c_1\right )+\cosh \left (c_1\right )\right ) \left (\sinh \left (c_1\right )+\cosh \left (c_1\right )+18 x\right ){}^3}-\sinh \left (c_1\right )-\cosh \left (c_1\right )+54 x}\right \},\left \{y(x)\to \frac {1}{3} \sqrt {\frac {2}{3}} \sqrt {\cosh \left (c_1\right ) \sqrt {\left (\sinh \left (c_1\right )+\cosh \left (c_1\right )\right ) \left (\sinh \left (c_1\right )+\cosh \left (c_1\right )+18 x\right ){}^3}-\sinh \left (c_1\right ) \sqrt {\left (\sinh \left (c_1\right )+\cosh \left (c_1\right )\right ) \left (\sinh \left (c_1\right )+\cosh \left (c_1\right )+18 x\right ){}^3}-\sinh \left (c_1\right )-\cosh \left (c_1\right )+54 x}\right \},\left \{y(x)\to -\frac {1}{3} \sqrt {\frac {2}{3}} \sqrt {-\cosh \left (c_1\right ) \sqrt {\left (\sinh \left (c_1\right )+\cosh \left (c_1\right )\right ) \left (\sinh \left (c_1\right )+\cosh \left (c_1\right )+18 x\right ){}^3}+\sinh \left (c_1\right ) \sqrt {\left (\sinh \left (c_1\right )+\cosh \left (c_1\right )\right ) \left (\sinh \left (c_1\right )+\cosh \left (c_1\right )+18 x\right ){}^3}-\sinh \left (c_1\right )-\cosh \left (c_1\right )+54 x}\right \},\left \{y(x)\to \frac {1}{3} \sqrt {\frac {2}{3}} \sqrt {-\cosh \left (c_1\right ) \sqrt {\left (\sinh \left (c_1\right )+\cosh \left (c_1\right )\right ) \left (\sinh \left (c_1\right )+\cosh \left (c_1\right )+18 x\right ){}^3}+\sinh \left (c_1\right ) \sqrt {\left (\sinh \left (c_1\right )+\cosh \left (c_1\right )\right ) \left (\sinh \left (c_1\right )+\cosh \left (c_1\right )+18 x\right ){}^3}-\sinh \left (c_1\right )-\cosh \left (c_1\right )+54 x}\right \}\right \}\]

Maple
cpu = 0.021 (sec), leaf count = 33

\[ \left \{ [x \left ({\it \_T} \right ) ={\frac {1}{{{\it \_T}}^{4}} \left ({\frac {3\,{{\it \_T}}^{2}}{2}}+{\it \_C1} \right ) },y \left ({\it \_T} \right ) ={\frac {9\,{{\it \_T}}^{2}+4\,{\it \_C1}}{3\,{{\it \_T}}^{3}}}] \right \} \] Mathematica raw input

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

Mathematica raw output

{{y[x] -> -(Sqrt[2/3]*Sqrt[54*x + Cosh[C[1]] + Sinh[C[1]] + (-Cosh[C[1]] + Sinh[
C[1]])*Sqrt[(Cosh[C[1]] + Sinh[C[1]])*(-18*x + Cosh[C[1]] + Sinh[C[1]])^3]])/3},
 {y[x] -> (Sqrt[2/3]*Sqrt[54*x + Cosh[C[1]] + Sinh[C[1]] + (-Cosh[C[1]] + Sinh[C
[1]])*Sqrt[(Cosh[C[1]] + Sinh[C[1]])*(-18*x + Cosh[C[1]] + Sinh[C[1]])^3]])/3}, 
{y[x] -> -(Sqrt[2/3]*Sqrt[54*x + Cosh[C[1]] + Sinh[C[1]] + (Cosh[C[1]] - Sinh[C[
1]])*Sqrt[(Cosh[C[1]] + Sinh[C[1]])*(-18*x + Cosh[C[1]] + Sinh[C[1]])^3]])/3}, {
y[x] -> (Sqrt[2/3]*Sqrt[54*x + Cosh[C[1]] + Sinh[C[1]] + (Cosh[C[1]] - Sinh[C[1]
])*Sqrt[(Cosh[C[1]] + Sinh[C[1]])*(-18*x + Cosh[C[1]] + Sinh[C[1]])^3]])/3}, {y[
x] -> -(Sqrt[2/3]*Sqrt[54*x - Cosh[C[1]] - Sinh[C[1]] + Cosh[C[1]]*Sqrt[(Cosh[C[
1]] + Sinh[C[1]])*(18*x + Cosh[C[1]] + Sinh[C[1]])^3] - Sinh[C[1]]*Sqrt[(Cosh[C[
1]] + Sinh[C[1]])*(18*x + Cosh[C[1]] + Sinh[C[1]])^3]])/3}, {y[x] -> (Sqrt[2/3]*
Sqrt[54*x - Cosh[C[1]] - Sinh[C[1]] + Cosh[C[1]]*Sqrt[(Cosh[C[1]] + Sinh[C[1]])*
(18*x + Cosh[C[1]] + Sinh[C[1]])^3] - Sinh[C[1]]*Sqrt[(Cosh[C[1]] + Sinh[C[1]])*
(18*x + Cosh[C[1]] + Sinh[C[1]])^3]])/3}, {y[x] -> -(Sqrt[2/3]*Sqrt[54*x - Cosh[
C[1]] - Sinh[C[1]] - Cosh[C[1]]*Sqrt[(Cosh[C[1]] + Sinh[C[1]])*(18*x + Cosh[C[1]
] + Sinh[C[1]])^3] + Sinh[C[1]]*Sqrt[(Cosh[C[1]] + Sinh[C[1]])*(18*x + Cosh[C[1]
] + Sinh[C[1]])^3]])/3}, {y[x] -> (Sqrt[2/3]*Sqrt[54*x - Cosh[C[1]] - Sinh[C[1]]
 - Cosh[C[1]]*Sqrt[(Cosh[C[1]] + Sinh[C[1]])*(18*x + Cosh[C[1]] + Sinh[C[1]])^3]
 + Sinh[C[1]]*Sqrt[(Cosh[C[1]] + Sinh[C[1]])*(18*x + Cosh[C[1]] + Sinh[C[1]])^3]
])/3}}

Maple raw input

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

Maple raw output

[x(_T) = 1/_T^4*(3/2*_T^2+_C1), y(_T) = 1/3*(9*_T^2+4*_C1)/_T^3]