4.17.44 \(5 y'(x)^2+3 x y'(x)-y(x)=0\)

ODE
\[ 5 y'(x)^2+3 x y'(x)-y(x)=0 \] ODE Classification

[[_1st_order, _with_linear_symmetries], _dAlembert]

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

Mathematica
cpu = 0.297902 (sec), leaf count = 771

\[\left \{\left \{y(x)\to \text {Root}\left [80 \text {$\#$1}^5+40 \text {$\#$1}^4 x^2+5 \text {$\#$1}^3 x^4-4000 \text {$\#$1}^2 e^{5 c_1} x-1800 \text {$\#$1} e^{5 c_1} x^3-216 e^{5 c_1} x^5-40000 e^{10 c_1}\& ,1\right ]\right \},\left \{y(x)\to \text {Root}\left [80 \text {$\#$1}^5+40 \text {$\#$1}^4 x^2+5 \text {$\#$1}^3 x^4-4000 \text {$\#$1}^2 e^{5 c_1} x-1800 \text {$\#$1} e^{5 c_1} x^3-216 e^{5 c_1} x^5-40000 e^{10 c_1}\& ,2\right ]\right \},\left \{y(x)\to \text {Root}\left [80 \text {$\#$1}^5+40 \text {$\#$1}^4 x^2+5 \text {$\#$1}^3 x^4-4000 \text {$\#$1}^2 e^{5 c_1} x-1800 \text {$\#$1} e^{5 c_1} x^3-216 e^{5 c_1} x^5-40000 e^{10 c_1}\& ,3\right ]\right \},\left \{y(x)\to \text {Root}\left [80 \text {$\#$1}^5+40 \text {$\#$1}^4 x^2+5 \text {$\#$1}^3 x^4-4000 \text {$\#$1}^2 e^{5 c_1} x-1800 \text {$\#$1} e^{5 c_1} x^3-216 e^{5 c_1} x^5-40000 e^{10 c_1}\& ,4\right ]\right \},\left \{y(x)\to \text {Root}\left [80 \text {$\#$1}^5+40 \text {$\#$1}^4 x^2+5 \text {$\#$1}^3 x^4-4000 \text {$\#$1}^2 e^{5 c_1} x-1800 \text {$\#$1} e^{5 c_1} x^3-216 e^{5 c_1} x^5-40000 e^{10 c_1}\& ,5\right ]\right \},\left \{y(x)\to \text {Root}\left [640000 \text {$\#$1}^5+320000 \text {$\#$1}^4 x^2+40000 \text {$\#$1}^3 x^4+4000 \text {$\#$1}^2 e^{5 c_1} x+1800 \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 [640000 \text {$\#$1}^5+320000 \text {$\#$1}^4 x^2+40000 \text {$\#$1}^3 x^4+4000 \text {$\#$1}^2 e^{5 c_1} x+1800 \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 [640000 \text {$\#$1}^5+320000 \text {$\#$1}^4 x^2+40000 \text {$\#$1}^3 x^4+4000 \text {$\#$1}^2 e^{5 c_1} x+1800 \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 [640000 \text {$\#$1}^5+320000 \text {$\#$1}^4 x^2+40000 \text {$\#$1}^3 x^4+4000 \text {$\#$1}^2 e^{5 c_1} x+1800 \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 [640000 \text {$\#$1}^5+320000 \text {$\#$1}^4 x^2+40000 \text {$\#$1}^3 x^4+4000 \text {$\#$1}^2 e^{5 c_1} x+1800 \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 = 0.018 (sec), leaf count = 32

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

DSolve[-y[x] + 3*x*y'[x] + 5*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> Root[-40000*E^(10*C[1]) - 216*E^(5*C[1])*x^5 - 1800*E^(5*C[1])*x^3*#1 
- 4000*E^(5*C[1])*x*#1^2 + 5*x^4*#1^3 + 40*x^2*#1^4 + 80*#1^5 & , 1]}, {y[x] -> 
Root[-40000*E^(10*C[1]) - 216*E^(5*C[1])*x^5 - 1800*E^(5*C[1])*x^3*#1 - 4000*E^(
5*C[1])*x*#1^2 + 5*x^4*#1^3 + 40*x^2*#1^4 + 80*#1^5 & , 2]}, {y[x] -> Root[-4000
0*E^(10*C[1]) - 216*E^(5*C[1])*x^5 - 1800*E^(5*C[1])*x^3*#1 - 4000*E^(5*C[1])*x*
#1^2 + 5*x^4*#1^3 + 40*x^2*#1^4 + 80*#1^5 & , 3]}, {y[x] -> Root[-40000*E^(10*C[
1]) - 216*E^(5*C[1])*x^5 - 1800*E^(5*C[1])*x^3*#1 - 4000*E^(5*C[1])*x*#1^2 + 5*x
^4*#1^3 + 40*x^2*#1^4 + 80*#1^5 & , 4]}, {y[x] -> Root[-40000*E^(10*C[1]) - 216*
E^(5*C[1])*x^5 - 1800*E^(5*C[1])*x^3*#1 - 4000*E^(5*C[1])*x*#1^2 + 5*x^4*#1^3 + 
40*x^2*#1^4 + 80*#1^5 & , 5]}, {y[x] -> Root[-5*E^(10*C[1]) + 216*E^(5*C[1])*x^5
 + 1800*E^(5*C[1])*x^3*#1 + 4000*E^(5*C[1])*x*#1^2 + 40000*x^4*#1^3 + 320000*x^2
*#1^4 + 640000*#1^5 & , 1]}, {y[x] -> Root[-5*E^(10*C[1]) + 216*E^(5*C[1])*x^5 +
 1800*E^(5*C[1])*x^3*#1 + 4000*E^(5*C[1])*x*#1^2 + 40000*x^4*#1^3 + 320000*x^2*#
1^4 + 640000*#1^5 & , 2]}, {y[x] -> Root[-5*E^(10*C[1]) + 216*E^(5*C[1])*x^5 + 1
800*E^(5*C[1])*x^3*#1 + 4000*E^(5*C[1])*x*#1^2 + 40000*x^4*#1^3 + 320000*x^2*#1^
4 + 640000*#1^5 & , 3]}, {y[x] -> Root[-5*E^(10*C[1]) + 216*E^(5*C[1])*x^5 + 180
0*E^(5*C[1])*x^3*#1 + 4000*E^(5*C[1])*x*#1^2 + 40000*x^4*#1^3 + 320000*x^2*#1^4 
+ 640000*#1^5 & , 4]}, {y[x] -> Root[-5*E^(10*C[1]) + 216*E^(5*C[1])*x^5 + 1800*
E^(5*C[1])*x^3*#1 + 4000*E^(5*C[1])*x*#1^2 + 40000*x^4*#1^3 + 320000*x^2*#1^4 + 
640000*#1^5 & , 5]}}

Maple raw input

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