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

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

[[_1st_order, _with_linear_symmetries], _dAlembert]

Book solution method
No Missing Variables ODE, Solve for \(y\)

Mathematica
cpu = 0.281866 (sec), leaf count = 771

\[\left \{\left \{y(x)\to \text {Root}\left [16 \text {$\#$1}^5+40 \text {$\#$1}^4 x^2+25 \text {$\#$1}^3 x^4-160 \text {$\#$1}^2 e^{5 c_1} x-360 \text {$\#$1} e^{5 c_1} x^3-216 e^{5 c_1} x^5-64 e^{10 c_1}\& ,1\right ]\right \},\left \{y(x)\to \text {Root}\left [16 \text {$\#$1}^5+40 \text {$\#$1}^4 x^2+25 \text {$\#$1}^3 x^4-160 \text {$\#$1}^2 e^{5 c_1} x-360 \text {$\#$1} e^{5 c_1} x^3-216 e^{5 c_1} x^5-64 e^{10 c_1}\& ,2\right ]\right \},\left \{y(x)\to \text {Root}\left [16 \text {$\#$1}^5+40 \text {$\#$1}^4 x^2+25 \text {$\#$1}^3 x^4-160 \text {$\#$1}^2 e^{5 c_1} x-360 \text {$\#$1} e^{5 c_1} x^3-216 e^{5 c_1} x^5-64 e^{10 c_1}\& ,3\right ]\right \},\left \{y(x)\to \text {Root}\left [16 \text {$\#$1}^5+40 \text {$\#$1}^4 x^2+25 \text {$\#$1}^3 x^4-160 \text {$\#$1}^2 e^{5 c_1} x-360 \text {$\#$1} e^{5 c_1} x^3-216 e^{5 c_1} x^5-64 e^{10 c_1}\& ,4\right ]\right \},\left \{y(x)\to \text {Root}\left [16 \text {$\#$1}^5+40 \text {$\#$1}^4 x^2+25 \text {$\#$1}^3 x^4-160 \text {$\#$1}^2 e^{5 c_1} x-360 \text {$\#$1} e^{5 c_1} x^3-216 e^{5 c_1} x^5-64 e^{10 c_1}\& ,5\right ]\right \},\left \{y(x)\to \text {Root}\left [1024 \text {$\#$1}^5+2560 \text {$\#$1}^4 x^2+1600 \text {$\#$1}^3 x^4+160 \text {$\#$1}^2 e^{5 c_1} x+360 \text {$\#$1} e^{5 c_1} x^3+216 e^{5 c_1} x^5-e^{10 c_1}\& ,1\right ]\right \},\left \{y(x)\to \text {Root}\left [1024 \text {$\#$1}^5+2560 \text {$\#$1}^4 x^2+1600 \text {$\#$1}^3 x^4+160 \text {$\#$1}^2 e^{5 c_1} x+360 \text {$\#$1} e^{5 c_1} x^3+216 e^{5 c_1} x^5-e^{10 c_1}\& ,2\right ]\right \},\left \{y(x)\to \text {Root}\left [1024 \text {$\#$1}^5+2560 \text {$\#$1}^4 x^2+1600 \text {$\#$1}^3 x^4+160 \text {$\#$1}^2 e^{5 c_1} x+360 \text {$\#$1} e^{5 c_1} x^3+216 e^{5 c_1} x^5-e^{10 c_1}\& ,3\right ]\right \},\left \{y(x)\to \text {Root}\left [1024 \text {$\#$1}^5+2560 \text {$\#$1}^4 x^2+1600 \text {$\#$1}^3 x^4+160 \text {$\#$1}^2 e^{5 c_1} x+360 \text {$\#$1} e^{5 c_1} x^3+216 e^{5 c_1} x^5-e^{10 c_1}\& ,4\right ]\right \},\left \{y(x)\to \text {Root}\left [1024 \text {$\#$1}^5+2560 \text {$\#$1}^4 x^2+1600 \text {$\#$1}^3 x^4+160 \text {$\#$1}^2 e^{5 c_1} x+360 \text {$\#$1} e^{5 c_1} x^3+216 e^{5 c_1} x^5-e^{10 c_1}\& ,5\right ]\right \}\right \}\]

Maple
cpu = 0.017 (sec), leaf count = 33

\[ \left \{ [x \left ({\it \_T} \right ) ={1 \left (-{\frac {2}{5}{{\it \_T}}^{{\frac {5}{2}}}}+{\it \_C1} \right ) {{\it \_T}}^{-{\frac {3}{2}}}},y \left ({\it \_T} \right ) ={\frac {1}{5} \left (-{{\it \_T}}^{{\frac {5}{2}}}+15\,{\it \_C1} \right ) {\frac {1}{\sqrt {{\it \_T}}}}}] \right \} \] Mathematica raw input

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

Mathematica raw output

{{y[x] -> Root[-64*E^(10*C[1]) - 216*E^(5*C[1])*x^5 - 360*E^(5*C[1])*x^3*#1 - 16
0*E^(5*C[1])*x*#1^2 + 25*x^4*#1^3 + 40*x^2*#1^4 + 16*#1^5 & , 1]}, {y[x] -> Root
[-64*E^(10*C[1]) - 216*E^(5*C[1])*x^5 - 360*E^(5*C[1])*x^3*#1 - 160*E^(5*C[1])*x
*#1^2 + 25*x^4*#1^3 + 40*x^2*#1^4 + 16*#1^5 & , 2]}, {y[x] -> Root[-64*E^(10*C[1
]) - 216*E^(5*C[1])*x^5 - 360*E^(5*C[1])*x^3*#1 - 160*E^(5*C[1])*x*#1^2 + 25*x^4
*#1^3 + 40*x^2*#1^4 + 16*#1^5 & , 3]}, {y[x] -> Root[-64*E^(10*C[1]) - 216*E^(5*
C[1])*x^5 - 360*E^(5*C[1])*x^3*#1 - 160*E^(5*C[1])*x*#1^2 + 25*x^4*#1^3 + 40*x^2
*#1^4 + 16*#1^5 & , 4]}, {y[x] -> Root[-64*E^(10*C[1]) - 216*E^(5*C[1])*x^5 - 36
0*E^(5*C[1])*x^3*#1 - 160*E^(5*C[1])*x*#1^2 + 25*x^4*#1^3 + 40*x^2*#1^4 + 16*#1^
5 & , 5]}, {y[x] -> Root[-E^(10*C[1]) + 216*E^(5*C[1])*x^5 + 360*E^(5*C[1])*x^3*
#1 + 160*E^(5*C[1])*x*#1^2 + 1600*x^4*#1^3 + 2560*x^2*#1^4 + 1024*#1^5 & , 1]}, 
{y[x] -> Root[-E^(10*C[1]) + 216*E^(5*C[1])*x^5 + 360*E^(5*C[1])*x^3*#1 + 160*E^
(5*C[1])*x*#1^2 + 1600*x^4*#1^3 + 2560*x^2*#1^4 + 1024*#1^5 & , 2]}, {y[x] -> Ro
ot[-E^(10*C[1]) + 216*E^(5*C[1])*x^5 + 360*E^(5*C[1])*x^3*#1 + 160*E^(5*C[1])*x*
#1^2 + 1600*x^4*#1^3 + 2560*x^2*#1^4 + 1024*#1^5 & , 3]}, {y[x] -> Root[-E^(10*C
[1]) + 216*E^(5*C[1])*x^5 + 360*E^(5*C[1])*x^3*#1 + 160*E^(5*C[1])*x*#1^2 + 1600
*x^4*#1^3 + 2560*x^2*#1^4 + 1024*#1^5 & , 4]}, {y[x] -> Root[-E^(10*C[1]) + 216*
E^(5*C[1])*x^5 + 360*E^(5*C[1])*x^3*#1 + 160*E^(5*C[1])*x*#1^2 + 1600*x^4*#1^3 +
 2560*x^2*#1^4 + 1024*#1^5 & , 5]}}

Maple raw input

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

Maple raw output

[x(_T) = 1/_T^(3/2)*(-2/5*_T^(5/2)+_C1), y(_T) = 1/5*(-_T^(5/2)+15*_C1)/_T^(1/2)
]