##### 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 = 0.473753 (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 = 0.047 (sec), leaf count = 85

$\left [\frac {\textit {\_C1}}{\left (-6 x -2 \sqrt {9 x^{2}+4 y \left (x \right )}\right )^{\frac {3}{2}}}+\frac {2 x}{5}-\frac {\sqrt {9 x^{2}+4 y \left (x \right )}}{5} = 0, \frac {\textit {\_C1}}{\left (-6 x +2 \sqrt {9 x^{2}+4 y \left (x \right )}\right )^{\frac {3}{2}}}+\frac {2 x}{5}+\frac {\sqrt {9 x^{2}+4 y \left (x \right )}}{5} = 0\right ]$ 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))

Maple raw output

[1/(-6*x-2*(9*x^2+4*y(x))^(1/2))^(3/2)*_C1+2/5*x-1/5*(9*x^2+4*y(x))^(1/2) = 0, 1
/(-6*x+2*(9*x^2+4*y(x))^(1/2))^(3/2)*_C1+2/5*x+1/5*(9*x^2+4*y(x))^(1/2) = 0]