##### 4.16.7 $$3 x^2+y'(x)^2=8 y(x)$$

ODE
$3 x^2+y'(x)^2=8 y(x)$ ODE Classiﬁcation

[[_homogeneous, class G]]

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

Mathematica
cpu = 0.926187 (sec), leaf count = 1865

$\left \{\left \{y(x)\to \frac {1}{96} \left (144 x^2+32 \cosh (2 c_1)+32 \sinh (2 c_1)-8\ 2^{2/3} \sqrt [3]{-729 \cosh (2 c_1) x^4-729 \sinh (2 c_1) x^4-270 \cosh (4 c_1) x^2-270 \sinh (4 c_1) x^2+2 \cosh (6 c_1)+2 \sinh (6 c_1)+3 \sqrt {3} \sqrt {x^2 \left (\left (27 x^2-4\right ) \cosh (c_1)-\left (27 x^2+4\right ) \sinh (c_1)\right ){}^3 (\cosh (7 c_1)+\sinh (7 c_1))}}-\frac {16 \sqrt [3]{2} \left (54 \cosh (2 c_1) x^2+54 \sinh (2 c_1) x^2+\cosh (4 c_1)+\sinh (4 c_1)\right )}{\sqrt [3]{-729 \cosh (2 c_1) x^4-729 \sinh (2 c_1) x^4-270 \cosh (4 c_1) x^2-270 \sinh (4 c_1) x^2+2 \cosh (6 c_1)+2 \sinh (6 c_1)+3 \sqrt {3} \sqrt {x^2 \left (\left (27 x^2-4\right ) \cosh (c_1)-\left (27 x^2+4\right ) \sinh (c_1)\right ){}^3 (\cosh (7 c_1)+\sinh (7 c_1))}}}\right )\right \},\left \{y(x)\to \frac {1}{192} \left (288 x^2+64 \cosh (2 c_1)+64 \sinh (2 c_1)+8\ 2^{2/3} \left (1-i \sqrt {3}\right ) \sqrt [3]{-729 \cosh (2 c_1) x^4-729 \sinh (2 c_1) x^4-270 \cosh (4 c_1) x^2-270 \sinh (4 c_1) x^2+2 \cosh (6 c_1)+2 \sinh (6 c_1)+3 \sqrt {3} \sqrt {x^2 \left (\left (27 x^2-4\right ) \cosh (c_1)-\left (27 x^2+4\right ) \sinh (c_1)\right ){}^3 (\cosh (7 c_1)+\sinh (7 c_1))}}+\frac {16 \sqrt [3]{2} \left (1+i \sqrt {3}\right ) \left (54 \cosh (2 c_1) x^2+54 \sinh (2 c_1) x^2+\cosh (4 c_1)+\sinh (4 c_1)\right )}{\sqrt [3]{-729 \cosh (2 c_1) x^4-729 \sinh (2 c_1) x^4-270 \cosh (4 c_1) x^2-270 \sinh (4 c_1) x^2+2 \cosh (6 c_1)+2 \sinh (6 c_1)+3 \sqrt {3} \sqrt {x^2 \left (\left (27 x^2-4\right ) \cosh (c_1)-\left (27 x^2+4\right ) \sinh (c_1)\right ){}^3 (\cosh (7 c_1)+\sinh (7 c_1))}}}\right )\right \},\left \{y(x)\to \frac {1}{192} \left (288 x^2+64 \cosh (2 c_1)+64 \sinh (2 c_1)+8\ 2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{-729 \cosh (2 c_1) x^4-729 \sinh (2 c_1) x^4-270 \cosh (4 c_1) x^2-270 \sinh (4 c_1) x^2+2 \cosh (6 c_1)+2 \sinh (6 c_1)+3 \sqrt {3} \sqrt {x^2 \left (\left (27 x^2-4\right ) \cosh (c_1)-\left (27 x^2+4\right ) \sinh (c_1)\right ){}^3 (\cosh (7 c_1)+\sinh (7 c_1))}}+\frac {16 \sqrt [3]{2} \left (1-i \sqrt {3}\right ) \left (54 \cosh (2 c_1) x^2+54 \sinh (2 c_1) x^2+\cosh (4 c_1)+\sinh (4 c_1)\right )}{\sqrt [3]{-729 \cosh (2 c_1) x^4-729 \sinh (2 c_1) x^4-270 \cosh (4 c_1) x^2-270 \sinh (4 c_1) x^2+2 \cosh (6 c_1)+2 \sinh (6 c_1)+3 \sqrt {3} \sqrt {x^2 \left (\left (27 x^2-4\right ) \cosh (c_1)-\left (27 x^2+4\right ) \sinh (c_1)\right ){}^3 (\cosh (7 c_1)+\sinh (7 c_1))}}}\right )\right \},\left \{y(x)\to \frac {1}{96} \left (144 x^2-32 \cosh (2 c_1)-32 \sinh (2 c_1)-8\ 2^{2/3} \sqrt [3]{729 \cosh (2 c_1) x^4+729 \sinh (2 c_1) x^4-270 \cosh (4 c_1) x^2-270 \sinh (4 c_1) x^2-2 \cosh (6 c_1)-2 \sinh (6 c_1)+3 \sqrt {3} \sqrt {x^2 \left (\left (27 x^2+4\right ) \cosh (c_1)+\left (4-27 x^2\right ) \sinh (c_1)\right ){}^3 (\cosh (7 c_1)+\sinh (7 c_1))}}+\frac {16 \sqrt [3]{2} \left (54 \cosh (2 c_1) x^2+54 \sinh (2 c_1) x^2-\cosh (4 c_1)-\sinh (4 c_1)\right )}{\sqrt [3]{729 \cosh (2 c_1) x^4+729 \sinh (2 c_1) x^4-270 \cosh (4 c_1) x^2-270 \sinh (4 c_1) x^2-2 \cosh (6 c_1)-2 \sinh (6 c_1)+3 \sqrt {3} \sqrt {x^2 \left (\left (27 x^2+4\right ) \cosh (c_1)+\left (4-27 x^2\right ) \sinh (c_1)\right ){}^3 (\cosh (7 c_1)+\sinh (7 c_1))}}}\right )\right \},\left \{y(x)\to \frac {1}{192} \left (288 x^2-64 \cosh (2 c_1)-64 \sinh (2 c_1)+8\ 2^{2/3} \left (1-i \sqrt {3}\right ) \sqrt [3]{729 \cosh (2 c_1) x^4+729 \sinh (2 c_1) x^4-270 \cosh (4 c_1) x^2-270 \sinh (4 c_1) x^2-2 \cosh (6 c_1)-2 \sinh (6 c_1)+3 \sqrt {3} \sqrt {x^2 \left (\left (27 x^2+4\right ) \cosh (c_1)+\left (4-27 x^2\right ) \sinh (c_1)\right ){}^3 (\cosh (7 c_1)+\sinh (7 c_1))}}-\frac {16 i \sqrt [3]{2} \left (-i+\sqrt {3}\right ) \left (54 \cosh (2 c_1) x^2+54 \sinh (2 c_1) x^2-\cosh (4 c_1)-\sinh (4 c_1)\right )}{\sqrt [3]{729 \cosh (2 c_1) x^4+729 \sinh (2 c_1) x^4-270 \cosh (4 c_1) x^2-270 \sinh (4 c_1) x^2-2 \cosh (6 c_1)-2 \sinh (6 c_1)+3 \sqrt {3} \sqrt {x^2 \left (\left (27 x^2+4\right ) \cosh (c_1)+\left (4-27 x^2\right ) \sinh (c_1)\right ){}^3 (\cosh (7 c_1)+\sinh (7 c_1))}}}\right )\right \},\left \{y(x)\to \frac {1}{192} \left (288 x^2-64 \cosh (2 c_1)-64 \sinh (2 c_1)+8\ 2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{729 \cosh (2 c_1) x^4+729 \sinh (2 c_1) x^4-270 \cosh (4 c_1) x^2-270 \sinh (4 c_1) x^2-2 \cosh (6 c_1)-2 \sinh (6 c_1)+3 \sqrt {3} \sqrt {x^2 \left (\left (27 x^2+4\right ) \cosh (c_1)+\left (4-27 x^2\right ) \sinh (c_1)\right ){}^3 (\cosh (7 c_1)+\sinh (7 c_1))}}+\frac {16 i \sqrt [3]{2} \left (i+\sqrt {3}\right ) \left (54 \cosh (2 c_1) x^2+54 \sinh (2 c_1) x^2-\cosh (4 c_1)-\sinh (4 c_1)\right )}{\sqrt [3]{729 \cosh (2 c_1) x^4+729 \sinh (2 c_1) x^4-270 \cosh (4 c_1) x^2-270 \sinh (4 c_1) x^2-2 \cosh (6 c_1)-2 \sinh (6 c_1)+3 \sqrt {3} \sqrt {x^2 \left (\left (27 x^2+4\right ) \cosh (c_1)+\left (4-27 x^2\right ) \sinh (c_1)\right ){}^3 (\cosh (7 c_1)+\sinh (7 c_1))}}}\right )\right \}\right \}$

Maple
cpu = 1.077 (sec), leaf count = 281

$\left [y \left (x \right ) = \frac {\RootOf \left (\textit {\_Z}^{18}-12 x \,\textit {\_Z}^{15}+60 x^{2} \textit {\_Z}^{12}-160 x^{3} \textit {\_Z}^{9}+\left (240 x^{4}-16 \textit {\_C1} \right ) \textit {\_Z}^{6}-192 x^{5} \textit {\_Z}^{3}+64 x^{6}\right )^{6}}{8}+\frac {x \RootOf \left (\textit {\_Z}^{18}-12 x \,\textit {\_Z}^{15}+60 x^{2} \textit {\_Z}^{12}-160 x^{3} \textit {\_Z}^{9}+\left (240 x^{4}-16 \textit {\_C1} \right ) \textit {\_Z}^{6}-192 x^{5} \textit {\_Z}^{3}+64 x^{6}\right )^{3}}{4}+\frac {x^{2}}{2}, y \left (x \right ) = \frac {\RootOf \left (\textit {\_C1} \,\textit {\_Z}^{18}+24 \textit {\_C1} x \,\textit {\_Z}^{15}+240 \textit {\_C1} \,x^{2} \textit {\_Z}^{12}+1280 \textit {\_C1} \,x^{3} \textit {\_Z}^{9}+\left (3840 \textit {\_C1} \,x^{4}-16\right ) \textit {\_Z}^{6}+\left (6144 \textit {\_C1} \,x^{5}-64 x \right ) \textit {\_Z}^{3}+4096 \textit {\_C1} \,x^{6}-64 x^{2}\right )^{6}}{8}+\frac {\RootOf \left (\textit {\_C1} \,\textit {\_Z}^{18}+24 \textit {\_C1} x \,\textit {\_Z}^{15}+240 \textit {\_C1} \,x^{2} \textit {\_Z}^{12}+1280 \textit {\_C1} \,x^{3} \textit {\_Z}^{9}+\left (3840 \textit {\_C1} \,x^{4}-16\right ) \textit {\_Z}^{6}+\left (6144 \textit {\_C1} \,x^{5}-64 x \right ) \textit {\_Z}^{3}+4096 \textit {\_C1} \,x^{6}-64 x^{2}\right )^{3} x}{4}+\frac {x^{2}}{2}\right ]$ Mathematica raw input

DSolve[3*x^2 + y'[x]^2 == 8*y[x],y[x],x]

Mathematica raw output

{{y[x] -> (144*x^2 + 32*Cosh[2*C[1]] + 32*Sinh[2*C[1]] - (16*2^(1/3)*(54*x^2*Cos
h[2*C[1]] + Cosh[4*C[1]] + 54*x^2*Sinh[2*C[1]] + Sinh[4*C[1]]))/(-729*x^4*Cosh[2
*C[1]] - 270*x^2*Cosh[4*C[1]] + 2*Cosh[6*C[1]] - 729*x^4*Sinh[2*C[1]] - 270*x^2*
Sinh[4*C[1]] + 2*Sinh[6*C[1]] + 3*Sqrt[3]*Sqrt[x^2*((-4 + 27*x^2)*Cosh[C[1]] - (
4 + 27*x^2)*Sinh[C[1]])^3*(Cosh[7*C[1]] + Sinh[7*C[1]])])^(1/3) - 8*2^(2/3)*(-72
9*x^4*Cosh[2*C[1]] - 270*x^2*Cosh[4*C[1]] + 2*Cosh[6*C[1]] - 729*x^4*Sinh[2*C[1]
] - 270*x^2*Sinh[4*C[1]] + 2*Sinh[6*C[1]] + 3*Sqrt[3]*Sqrt[x^2*((-4 + 27*x^2)*Co
sh[C[1]] - (4 + 27*x^2)*Sinh[C[1]])^3*(Cosh[7*C[1]] + Sinh[7*C[1]])])^(1/3))/96}
, {y[x] -> (288*x^2 + 64*Cosh[2*C[1]] + 64*Sinh[2*C[1]] + (16*2^(1/3)*(1 + I*Sqr
t[3])*(54*x^2*Cosh[2*C[1]] + Cosh[4*C[1]] + 54*x^2*Sinh[2*C[1]] + Sinh[4*C[1]]))
/(-729*x^4*Cosh[2*C[1]] - 270*x^2*Cosh[4*C[1]] + 2*Cosh[6*C[1]] - 729*x^4*Sinh[2
*C[1]] - 270*x^2*Sinh[4*C[1]] + 2*Sinh[6*C[1]] + 3*Sqrt[3]*Sqrt[x^2*((-4 + 27*x^
2)*Cosh[C[1]] - (4 + 27*x^2)*Sinh[C[1]])^3*(Cosh[7*C[1]] + Sinh[7*C[1]])])^(1/3)
+ 8*2^(2/3)*(1 - I*Sqrt[3])*(-729*x^4*Cosh[2*C[1]] - 270*x^2*Cosh[4*C[1]] + 2*C
osh[6*C[1]] - 729*x^4*Sinh[2*C[1]] - 270*x^2*Sinh[4*C[1]] + 2*Sinh[6*C[1]] + 3*S
qrt[3]*Sqrt[x^2*((-4 + 27*x^2)*Cosh[C[1]] - (4 + 27*x^2)*Sinh[C[1]])^3*(Cosh[7*C
[1]] + Sinh[7*C[1]])])^(1/3))/192}, {y[x] -> (288*x^2 + 64*Cosh[2*C[1]] + 64*Sin
h[2*C[1]] + (16*2^(1/3)*(1 - I*Sqrt[3])*(54*x^2*Cosh[2*C[1]] + Cosh[4*C[1]] + 54
*x^2*Sinh[2*C[1]] + Sinh[4*C[1]]))/(-729*x^4*Cosh[2*C[1]] - 270*x^2*Cosh[4*C[1]]
+ 2*Cosh[6*C[1]] - 729*x^4*Sinh[2*C[1]] - 270*x^2*Sinh[4*C[1]] + 2*Sinh[6*C[1]]
+ 3*Sqrt[3]*Sqrt[x^2*((-4 + 27*x^2)*Cosh[C[1]] - (4 + 27*x^2)*Sinh[C[1]])^3*(Co
sh[7*C[1]] + Sinh[7*C[1]])])^(1/3) + 8*2^(2/3)*(1 + I*Sqrt[3])*(-729*x^4*Cosh[2*
C[1]] - 270*x^2*Cosh[4*C[1]] + 2*Cosh[6*C[1]] - 729*x^4*Sinh[2*C[1]] - 270*x^2*S
inh[4*C[1]] + 2*Sinh[6*C[1]] + 3*Sqrt[3]*Sqrt[x^2*((-4 + 27*x^2)*Cosh[C[1]] - (4
+ 27*x^2)*Sinh[C[1]])^3*(Cosh[7*C[1]] + Sinh[7*C[1]])])^(1/3))/192}, {y[x] -> (
144*x^2 - 32*Cosh[2*C[1]] - 32*Sinh[2*C[1]] + (16*2^(1/3)*(54*x^2*Cosh[2*C[1]] -
Cosh[4*C[1]] + 54*x^2*Sinh[2*C[1]] - Sinh[4*C[1]]))/(729*x^4*Cosh[2*C[1]] - 270
*x^2*Cosh[4*C[1]] - 2*Cosh[6*C[1]] + 729*x^4*Sinh[2*C[1]] - 270*x^2*Sinh[4*C[1]]
- 2*Sinh[6*C[1]] + 3*Sqrt[3]*Sqrt[x^2*((4 + 27*x^2)*Cosh[C[1]] + (4 - 27*x^2)*S
inh[C[1]])^3*(Cosh[7*C[1]] + Sinh[7*C[1]])])^(1/3) - 8*2^(2/3)*(729*x^4*Cosh[2*C
[1]] - 270*x^2*Cosh[4*C[1]] - 2*Cosh[6*C[1]] + 729*x^4*Sinh[2*C[1]] - 270*x^2*Si
nh[4*C[1]] - 2*Sinh[6*C[1]] + 3*Sqrt[3]*Sqrt[x^2*((4 + 27*x^2)*Cosh[C[1]] + (4 -
27*x^2)*Sinh[C[1]])^3*(Cosh[7*C[1]] + Sinh[7*C[1]])])^(1/3))/96}, {y[x] -> (288
*x^2 - 64*Cosh[2*C[1]] - 64*Sinh[2*C[1]] - ((16*I)*2^(1/3)*(-I + Sqrt[3])*(54*x^
2*Cosh[2*C[1]] - Cosh[4*C[1]] + 54*x^2*Sinh[2*C[1]] - Sinh[4*C[1]]))/(729*x^4*Co
sh[2*C[1]] - 270*x^2*Cosh[4*C[1]] - 2*Cosh[6*C[1]] + 729*x^4*Sinh[2*C[1]] - 270*
x^2*Sinh[4*C[1]] - 2*Sinh[6*C[1]] + 3*Sqrt[3]*Sqrt[x^2*((4 + 27*x^2)*Cosh[C[1]]
+ (4 - 27*x^2)*Sinh[C[1]])^3*(Cosh[7*C[1]] + Sinh[7*C[1]])])^(1/3) + 8*2^(2/3)*(
1 - I*Sqrt[3])*(729*x^4*Cosh[2*C[1]] - 270*x^2*Cosh[4*C[1]] - 2*Cosh[6*C[1]] + 7
29*x^4*Sinh[2*C[1]] - 270*x^2*Sinh[4*C[1]] - 2*Sinh[6*C[1]] + 3*Sqrt[3]*Sqrt[x^2
*((4 + 27*x^2)*Cosh[C[1]] + (4 - 27*x^2)*Sinh[C[1]])^3*(Cosh[7*C[1]] + Sinh[7*C[
1]])])^(1/3))/192}, {y[x] -> (288*x^2 - 64*Cosh[2*C[1]] - 64*Sinh[2*C[1]] + ((16
*I)*2^(1/3)*(I + Sqrt[3])*(54*x^2*Cosh[2*C[1]] - Cosh[4*C[1]] + 54*x^2*Sinh[2*C[
1]] - Sinh[4*C[1]]))/(729*x^4*Cosh[2*C[1]] - 270*x^2*Cosh[4*C[1]] - 2*Cosh[6*C[1
]] + 729*x^4*Sinh[2*C[1]] - 270*x^2*Sinh[4*C[1]] - 2*Sinh[6*C[1]] + 3*Sqrt[3]*Sq
rt[x^2*((4 + 27*x^2)*Cosh[C[1]] + (4 - 27*x^2)*Sinh[C[1]])^3*(Cosh[7*C[1]] + Sin
h[7*C[1]])])^(1/3) + 8*2^(2/3)*(1 + I*Sqrt[3])*(729*x^4*Cosh[2*C[1]] - 270*x^2*C
osh[4*C[1]] - 2*Cosh[6*C[1]] + 729*x^4*Sinh[2*C[1]] - 270*x^2*Sinh[4*C[1]] - 2*S
inh[6*C[1]] + 3*Sqrt[3]*Sqrt[x^2*((4 + 27*x^2)*Cosh[C[1]] + (4 - 27*x^2)*Sinh[C[
1]])^3*(Cosh[7*C[1]] + Sinh[7*C[1]])])^(1/3))/192}}

Maple raw input

dsolve(diff(y(x),x)^2+3*x^2 = 8*y(x), y(x))

Maple raw output

[y(x) = 1/8*RootOf(_Z^18-12*x*_Z^15+60*x^2*_Z^12-160*x^3*_Z^9+(240*x^4-16*_C1)*_
Z^6-192*x^5*_Z^3+64*x^6)^6+1/4*x*RootOf(_Z^18-12*x*_Z^15+60*x^2*_Z^12-160*x^3*_Z
^9+(240*x^4-16*_C1)*_Z^6-192*x^5*_Z^3+64*x^6)^3+1/2*x^2, y(x) = 1/8*RootOf(_C1*_
Z^18+24*_C1*x*_Z^15+240*_C1*x^2*_Z^12+1280*_C1*x^3*_Z^9+(3840*_C1*x^4-16)*_Z^6+(
6144*_C1*x^5-64*x)*_Z^3+4096*_C1*x^6-64*x^2)^6+1/4*RootOf(_C1*_Z^18+24*_C1*x*_Z^
15+240*_C1*x^2*_Z^12+1280*_C1*x^3*_Z^9+(3840*_C1*x^4-16)*_Z^6+(6144*_C1*x^5-64*x
)*_Z^3+4096*_C1*x^6-64*x^2)^3*x+1/2*x^2]