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 = 0.740935 (sec), leaf count = 1865

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

Maple
cpu = 0.137 (sec), leaf count = 418

\[ \left \{ {\frac {1}{ \left (3\,{x}^{2}-2\,y \relax (x ) \right ) ^{3}} \left (\left (3\,{x}^{5}-14\,{x}^{3}y \relax (x ) +16\,x \left (y \relax (x ) \right ) ^{2} \right ) \sqrt {8\,y \relax (x ) -3\,{x}^{2}}-512\,{\it \_C1}\, \left (y \relax (x ) \right ) ^{5}+1536\,{x}^{2}{\it \_C1}\, \left (y \relax (x ) \right ) ^{4}+ \left (64\, \left (8\,y \relax (x ) -3\,{x}^{2} \right ) ^{3/2}x{\it \_C1}+576\,{x}^{4}{\it \_C1}+16 \right ) \left (y \relax (x ) \right ) ^{3}+ \left (-288\, \left (8\,y \relax (x ) -3\,{x}^{2} \right ) ^{3/2}{x}^{3}{\it \_C1}-6048\,{\it \_C1}\,{x}^{6}+16\,{x}^{2} \right ) \left (y \relax (x ) \right ) ^{2}+ \left (432\, \left (8\,y \relax (x ) -3\,{x}^{2} \right ) ^{3/2}{x}^{5}{\it \_C1}+6480\,{\it \_C1}\,{x}^{8}-30\,{x}^{4} \right ) y \relax (x ) -216\, \left (8\,y \relax (x ) -3\,{x}^{2} \right ) ^{3/2}{\it \_C1}\,{x}^{7}-1944\,{\it \_C1}\,{x}^{10}+9\,{x}^{6} \right ) \left (3\,x-\sqrt {8\,y \relax (x ) -3\,{x}^{2}} \right ) ^{-3} \left (\sqrt {8\,y \relax (x ) -3\,{x}^{2}}+x \right ) ^{-1}}=0,{\frac {1}{{x}^{2}-2\,y \relax (x ) } \left (81\,x \left ({x}^{2}-8/3\,y \relax (x ) \right ) \left ({x}^{2}-2/3\,y \relax (x ) \right ) ^{3}\sqrt {8\,y \relax (x ) -3\,{x}^{2}}+64\, \left (y \relax (x ) \right ) ^{5}-192\,{x}^{2} \left (y \relax (x ) \right ) ^{4}+ \left (-72\,{x}^{4}-128\,{\it \_C1} \right ) \left (y \relax (x ) \right ) ^{3}+ \left (756\,{x}^{6}-128\,{\it \_C1}\,{x}^{2} \right ) \left (y \relax (x ) \right ) ^{2}+ \left (-810\,{x}^{8}+16\, \left (8\,y \relax (x ) -3\,{x}^{2} \right ) ^{3/2}x{\it \_C1}+240\,{x}^{4}{\it \_C1} \right ) y \relax (x ) +243\,{x}^{10}-8\, \left (8\,y \relax (x ) -3\,{x}^{2} \right ) ^{3/2}{x}^{3}{\it \_C1}-72\,{\it \_C1}\,{x}^{6} \right ) \left (3\,x-\sqrt {8\,y \relax (x ) -3\,{x}^{2}} \right ) ^{-3} \left (\sqrt {8\,y \relax (x ) -3\,{x}^{2}}+x \right ) ^{-1}}=0 \right \} \] 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),'implicit')

Maple raw output

((3*x^5-14*x^3*y(x)+16*x*y(x)^2)*(8*y(x)-3*x^2)^(1/2)-512*_C1*y(x)^5+1536*x^2*_C
1*y(x)^4+(64*(8*y(x)-3*x^2)^(3/2)*x*_C1+576*x^4*_C1+16)*y(x)^3+(-288*(8*y(x)-3*x
^2)^(3/2)*x^3*_C1-6048*_C1*x^6+16*x^2)*y(x)^2+(432*(8*y(x)-3*x^2)^(3/2)*x^5*_C1+
6480*_C1*x^8-30*x^4)*y(x)-216*(8*y(x)-3*x^2)^(3/2)*_C1*x^7-1944*_C1*x^10+9*x^6)/
(3*x-(8*y(x)-3*x^2)^(1/2))^3/((8*y(x)-3*x^2)^(1/2)+x)/(3*x^2-2*y(x))^3 = 0, (81*
x*(x^2-8/3*y(x))*(x^2-2/3*y(x))^3*(8*y(x)-3*x^2)^(1/2)+64*y(x)^5-192*x^2*y(x)^4+
(-72*x^4-128*_C1)*y(x)^3+(756*x^6-128*_C1*x^2)*y(x)^2+(-810*x^8+16*(8*y(x)-3*x^2
)^(3/2)*x*_C1+240*x^4*_C1)*y(x)+243*x^10-8*(8*y(x)-3*x^2)^(3/2)*x^3*_C1-72*_C1*x
^6)/(3*x-(8*y(x)-3*x^2)^(1/2))^3/((8*y(x)-3*x^2)^(1/2)+x)/(x^2-2*y(x)) = 0