ODE
\[ y(x)^5 \left (-y'(x)\right )+3 x y(x)^4 y'(x)^2+1=0 \] ODE Classification
[[_homogeneous, `class G`], _rational]
Book solution method
Change of variable
Mathematica ✓
cpu = 0.390326 (sec), leaf count = 408
\[\left \{\left \{y(x)\to -\sqrt [3]{-\frac {1}{2}} \sqrt [3]{-e^{-\frac {c_1}{2}} \left (e^{c_1}+12 x\right )}\right \},\left \{y(x)\to \frac {\sqrt [3]{-e^{-\frac {c_1}{2}} \left (e^{c_1}+12 x\right )}}{\sqrt [3]{2}}\right \},\left \{y(x)\to \frac {(-1)^{2/3} \sqrt [3]{-e^{-\frac {c_1}{2}} \left (e^{c_1}+12 x\right )}}{\sqrt [3]{2}}\right \},\left \{y(x)\to -\frac {\sqrt [6]{\left (144 x^2-1\right ) \sinh \left (c_1\right )-\left (144 x^2+1\right ) \cosh \left (c_1\right )+24 x}}{\sqrt [3]{2}}\right \},\left \{y(x)\to \frac {\sqrt [6]{\left (144 x^2-1\right ) \sinh \left (c_1\right )-\left (144 x^2+1\right ) \cosh \left (c_1\right )+24 x}}{\sqrt [3]{2}}\right \},\left \{y(x)\to \frac {\left (1-i \sqrt {3}\right ) \sqrt [6]{\left (144 x^2-1\right ) \sinh \left (c_1\right )-\left (144 x^2+1\right ) \cosh \left (c_1\right )+24 x}}{2 \sqrt [3]{2}}\right \},\left \{y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [6]{\left (144 x^2-1\right ) \sinh \left (c_1\right )-\left (144 x^2+1\right ) \cosh \left (c_1\right )+24 x}}{2 \sqrt [3]{2}}\right \},\left \{y(x)\to -\frac {i \left (\sqrt {3}-i\right ) \sqrt [6]{\left (144 x^2-1\right ) \sinh \left (c_1\right )-\left (144 x^2+1\right ) \cosh \left (c_1\right )+24 x}}{2 \sqrt [3]{2}}\right \},\left \{y(x)\to \frac {\left (1+i \sqrt {3}\right ) \sqrt [6]{\left (144 x^2-1\right ) \sinh \left (c_1\right )-\left (144 x^2+1\right ) \cosh \left (c_1\right )+24 x}}{2 \sqrt [3]{2}}\right \}\right \}\]
Maple ✓
cpu = 0.146 (sec), leaf count = 99
\[ \left \{ \left ( y \left ( x \right ) \right ) ^{6}-12\,x=0,-\ln \left ( x \right ) -\ln \left ( - \left ( y \left ( x \right ) \right ) ^{3}+\sqrt { \left ( y \left ( x \right ) \right ) ^{6}-12\,x} \right ) +\ln \left ( \left ( y \left ( x \right ) \right ) ^{3}+\sqrt { \left ( y \left ( x \right ) \right ) ^{6}-12\,x} \right ) +{\it \_C1}=0,-\ln \left ( x \right ) +\ln \left ( - \left ( y \left ( x \right ) \right ) ^{3}+\sqrt { \left ( y \left ( x \right ) \right ) ^{6}-12\,x} \right ) -\ln \left ( \left ( y \left ( x \right ) \right ) ^{3}+\sqrt { \left ( y \left ( x \right ) \right ) ^{6}-12\,x} \right ) +{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[1 - y[x]^5*y'[x] + 3*x*y[x]^4*y'[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> -((-1/2)^(1/3)*(-((E^C[1] + 12*x)/E^(C[1]/2)))^(1/3))}, {y[x] -> (-((E^C[1] + 12*x)/E^(C[1]/2)))^(1/3)/2^(1/3)}, {y[x] -> ((-1)^(2/3)*(-((E^C[1] + 12*x)/E^(C[1]/2)))^(1/3))/2^(1/3)}, {y[x] -> -((24*x - (1 + 144*x^2)*Cosh[C[1]] + (-1 + 144*x^2)*Sinh[C[1]])^(1/6)/2^(1/3))}, {y[x] -> (24*x - (1 + 144*x^2)*Cosh[C[1]] + (-1 + 144*x^2)*Sinh[C[1]])^(1/6)/2^(1/3)}, {y[x] -> ((1 - I*Sqrt[3])*(24*x - (1 + 144*x^2)*Cosh[C[1]] + (-1 + 144*x^2)*Sinh[C[1]])^(1/6))/(2*2^(1/3))}, {y[x] -> ((I/2)*(I + Sqrt[3])*(24*x - (1 + 144*x^2)*Cosh[C[1]] + (-1 + 144*x^2)*Sinh[C[1]])^(1/6))/2^(1/3)}, {y[x] -> ((-I/2)*(-I + Sqrt[3])*(24*x - (1 + 144*x^2)*Cosh[C[1]] + (-1 + 144*x^2)*Sinh[C[1]])^(1/6))/2^(1/3)}, {y[x] -> ((1 + I*Sqrt[3])*(24*x - (1 + 144*x^2)*Cosh[C[1]] + (-1 + 144*x^2)*Sinh[C[1]])^(1/6))/(2*2^(1/3))}}
Maple raw input
dsolve(3*x*y(x)^4*diff(y(x),x)^2-y(x)^5*diff(y(x),x)+1 = 0, y(x),'implicit')
Maple raw output
y(x)^6-12*x = 0, -ln(x)+ln(-y(x)^3+(y(x)^6-12*x)^(1/2))-ln(y(x)^3+(y(x)^6-12*x)^(1/2))+_C1 = 0, -ln(x)-ln(-y(x)^3+(y(x)^6-12*x)^(1/2))+ln(y(x)^3+(y(x)^6-12*x)^(1/2))+_C1 = 0