4.17.45 \(5 y'(x)^2+6 x y'(x)-2 y(x)=0\)

ODE
\[ 5 y'(x)^2+6 x y'(x)-2 y(x)=0 \] ODE Classification

[[_1st_order, _with_linear_symmetries], _dAlembert]

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

Mathematica
cpu = 0.29187 (sec), leaf count = 771

\[\left \{\left \{y(x)\to \text {Root}\left [20 \text {$\#$1}^5+20 \text {$\#$1}^4 x^2+5 \text {$\#$1}^3 x^4-1000 \text {$\#$1}^2 e^{5 c_1} x-900 \text {$\#$1} e^{5 c_1} x^3-216 e^{5 c_1} x^5-5000 e^{10 c_1}\& ,1\right ]\right \},\left \{y(x)\to \text {Root}\left [20 \text {$\#$1}^5+20 \text {$\#$1}^4 x^2+5 \text {$\#$1}^3 x^4-1000 \text {$\#$1}^2 e^{5 c_1} x-900 \text {$\#$1} e^{5 c_1} x^3-216 e^{5 c_1} x^5-5000 e^{10 c_1}\& ,2\right ]\right \},\left \{y(x)\to \text {Root}\left [20 \text {$\#$1}^5+20 \text {$\#$1}^4 x^2+5 \text {$\#$1}^3 x^4-1000 \text {$\#$1}^2 e^{5 c_1} x-900 \text {$\#$1} e^{5 c_1} x^3-216 e^{5 c_1} x^5-5000 e^{10 c_1}\& ,3\right ]\right \},\left \{y(x)\to \text {Root}\left [20 \text {$\#$1}^5+20 \text {$\#$1}^4 x^2+5 \text {$\#$1}^3 x^4-1000 \text {$\#$1}^2 e^{5 c_1} x-900 \text {$\#$1} e^{5 c_1} x^3-216 e^{5 c_1} x^5-5000 e^{10 c_1}\& ,4\right ]\right \},\left \{y(x)\to \text {Root}\left [20 \text {$\#$1}^5+20 \text {$\#$1}^4 x^2+5 \text {$\#$1}^3 x^4-1000 \text {$\#$1}^2 e^{5 c_1} x-900 \text {$\#$1} e^{5 c_1} x^3-216 e^{5 c_1} x^5-5000 e^{10 c_1}\& ,5\right ]\right \},\left \{y(x)\to \text {Root}\left [20000 \text {$\#$1}^5+20000 \text {$\#$1}^4 x^2+5000 \text {$\#$1}^3 x^4+1000 \text {$\#$1}^2 e^{5 c_1} x+900 \text {$\#$1} e^{5 c_1} x^3+216 e^{5 c_1} x^5-5 e^{10 c_1}\& ,1\right ]\right \},\left \{y(x)\to \text {Root}\left [20000 \text {$\#$1}^5+20000 \text {$\#$1}^4 x^2+5000 \text {$\#$1}^3 x^4+1000 \text {$\#$1}^2 e^{5 c_1} x+900 \text {$\#$1} e^{5 c_1} x^3+216 e^{5 c_1} x^5-5 e^{10 c_1}\& ,2\right ]\right \},\left \{y(x)\to \text {Root}\left [20000 \text {$\#$1}^5+20000 \text {$\#$1}^4 x^2+5000 \text {$\#$1}^3 x^4+1000 \text {$\#$1}^2 e^{5 c_1} x+900 \text {$\#$1} e^{5 c_1} x^3+216 e^{5 c_1} x^5-5 e^{10 c_1}\& ,3\right ]\right \},\left \{y(x)\to \text {Root}\left [20000 \text {$\#$1}^5+20000 \text {$\#$1}^4 x^2+5000 \text {$\#$1}^3 x^4+1000 \text {$\#$1}^2 e^{5 c_1} x+900 \text {$\#$1} e^{5 c_1} x^3+216 e^{5 c_1} x^5-5 e^{10 c_1}\& ,4\right ]\right \},\left \{y(x)\to \text {Root}\left [20000 \text {$\#$1}^5+20000 \text {$\#$1}^4 x^2+5000 \text {$\#$1}^3 x^4+1000 \text {$\#$1}^2 e^{5 c_1} x+900 \text {$\#$1} e^{5 c_1} x^3+216 e^{5 c_1} x^5-5 e^{10 c_1}\& ,5\right ]\right \}\right \}\]

Maple
cpu = 0.016 (sec), leaf count = 33

\[ \left \{ [x \left ({\it \_T} \right ) ={1 \left (-{{\it \_T}}^{{\frac {5}{2}}}+{\it \_C1} \right ) {{\it \_T}}^{-{\frac {3}{2}}}},y \left ({\it \_T} \right ) ={\frac {1}{2} \left (-{{\it \_T}}^{{\frac {5}{2}}}+6\,{\it \_C1} \right ) {\frac {1}{\sqrt {{\it \_T}}}}}] \right \} \] Mathematica raw input

DSolve[-2*y[x] + 6*x*y'[x] + 5*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> Root[-5000*E^(10*C[1]) - 216*E^(5*C[1])*x^5 - 900*E^(5*C[1])*x^3*#1 - 
1000*E^(5*C[1])*x*#1^2 + 5*x^4*#1^3 + 20*x^2*#1^4 + 20*#1^5 & , 1]}, {y[x] -> Ro
ot[-5000*E^(10*C[1]) - 216*E^(5*C[1])*x^5 - 900*E^(5*C[1])*x^3*#1 - 1000*E^(5*C[
1])*x*#1^2 + 5*x^4*#1^3 + 20*x^2*#1^4 + 20*#1^5 & , 2]}, {y[x] -> Root[-5000*E^(
10*C[1]) - 216*E^(5*C[1])*x^5 - 900*E^(5*C[1])*x^3*#1 - 1000*E^(5*C[1])*x*#1^2 +
 5*x^4*#1^3 + 20*x^2*#1^4 + 20*#1^5 & , 3]}, {y[x] -> Root[-5000*E^(10*C[1]) - 2
16*E^(5*C[1])*x^5 - 900*E^(5*C[1])*x^3*#1 - 1000*E^(5*C[1])*x*#1^2 + 5*x^4*#1^3 
+ 20*x^2*#1^4 + 20*#1^5 & , 4]}, {y[x] -> Root[-5000*E^(10*C[1]) - 216*E^(5*C[1]
)*x^5 - 900*E^(5*C[1])*x^3*#1 - 1000*E^(5*C[1])*x*#1^2 + 5*x^4*#1^3 + 20*x^2*#1^
4 + 20*#1^5 & , 5]}, {y[x] -> Root[-5*E^(10*C[1]) + 216*E^(5*C[1])*x^5 + 900*E^(
5*C[1])*x^3*#1 + 1000*E^(5*C[1])*x*#1^2 + 5000*x^4*#1^3 + 20000*x^2*#1^4 + 20000
*#1^5 & , 1]}, {y[x] -> Root[-5*E^(10*C[1]) + 216*E^(5*C[1])*x^5 + 900*E^(5*C[1]
)*x^3*#1 + 1000*E^(5*C[1])*x*#1^2 + 5000*x^4*#1^3 + 20000*x^2*#1^4 + 20000*#1^5 
& , 2]}, {y[x] -> Root[-5*E^(10*C[1]) + 216*E^(5*C[1])*x^5 + 900*E^(5*C[1])*x^3*
#1 + 1000*E^(5*C[1])*x*#1^2 + 5000*x^4*#1^3 + 20000*x^2*#1^4 + 20000*#1^5 & , 3]
}, {y[x] -> Root[-5*E^(10*C[1]) + 216*E^(5*C[1])*x^5 + 900*E^(5*C[1])*x^3*#1 + 1
000*E^(5*C[1])*x*#1^2 + 5000*x^4*#1^3 + 20000*x^2*#1^4 + 20000*#1^5 & , 4]}, {y[
x] -> Root[-5*E^(10*C[1]) + 216*E^(5*C[1])*x^5 + 900*E^(5*C[1])*x^3*#1 + 1000*E^
(5*C[1])*x*#1^2 + 5000*x^4*#1^3 + 20000*x^2*#1^4 + 20000*#1^5 & , 5]}}

Maple raw input

dsolve(5*diff(y(x),x)^2+6*x*diff(y(x),x)-2*y(x) = 0, y(x),'implicit')

Maple raw output

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