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 = 0.496629 (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 = 0.113 (sec), leaf count = 85
\[\left [\frac {\textit {\_C1}}{\left (-30 x -10 \sqrt {9 x^{2}+20 y \left (x \right )}\right )^{\frac {3}{2}}}+\frac {2 x}{5}-\frac {\sqrt {9 x^{2}+20 y \left (x \right )}}{5} = 0, \frac {\textit {\_C1}}{\left (-30 x +10 \sqrt {9 x^{2}+20 y \left (x \right )}\right )^{\frac {3}{2}}}+\frac {2 x}{5}+\frac {\sqrt {9 x^{2}+20 y \left (x \right )}}{5} = 0\right ]\] 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[-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 & , 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 + 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 & , 3]}, {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 & , 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))
Maple raw output
[1/(-30*x-10*(9*x^2+20*y(x))^(1/2))^(3/2)*_C1+2/5*x-1/5*(9*x^2+20*y(x))^(1/2) = 0, 1/(-30*x+10*(9*x^2+20*y(x))^(1/2))^(3/2)*_C1+2/5*x+1/5*(9*x^2+20*y(x))^(1/2) = 0]