4.18.40 $4x{y}^{\prime }(x{)}^2-3y(x)y^{\prime }(x)+3=0$

ODE
$$4x{y}^{\prime }(x{)}^2-3y(x)y^{\prime }(x)+3=0$$ 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

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

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

$$\{[x\left(\mathit{\_}\mathit{T}\right)=\frac{1}{{\mathit{\_}\mathit{T}}^4}(\frac{3{\mathit{\_}\mathit{T}}^2}{2}+\mathit{\_}\mathit{C}\mathit{1}),y\left(\mathit{\_}\mathit{T}\right)=\frac{9{\mathit{\_}\mathit{T}}^2+4\mathit{\_}\mathit{C}\mathit{1}}{3{\mathit{\_}\mathit{T}}^3}]\}$$ 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]