##### 4.18.7 $$a+x y'(x)^2+y(x) y'(x)=0$$

ODE
$a+x y'(x)^2+y(x) y'(x)=0$ ODE Classiﬁcation

[[_homogeneous, class G], _dAlembert]

Book solution method
Clairaut’s equation and related types, d’Alembert’s equation (also call Lagrange’s)

Mathematica
cpu = 0.604601 (sec), leaf count = 2419

$\left \{\left \{y(x)\to \frac {-a \sqrt {\frac {-2304 a^3 x^3+36 \sqrt [3]{2} a \sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {e^{3 c_1} \left (e^{3 c_1}-1024 a^3 x^3\right ){}^3}} x+\frac {576\ 2^{2/3} a \left (128 a^4 x^4+a e^{3 c_1} x\right ) x}{\sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {e^{3 c_1} \left (e^{3 c_1}-1024 a^3 x^3\right ){}^3}}}+e^{3 c_1}}{a^2 x^2}} x-a \sqrt {-\frac {2 e^{\frac {3 c_1}{2}} \left (27648+\frac {e^{3 c_1}}{a^3 x^3}\right )}{\sqrt {\frac {-2304 a^3 x^3+36 \sqrt [3]{2} a \sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {e^{3 c_1} \left (e^{3 c_1}-1024 a^3 x^3\right ){}^3}} x+\frac {576\ 2^{2/3} a \left (128 a^4 x^4+a e^{3 c_1} x\right ) x}{\sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {e^{3 c_1} \left (e^{3 c_1}-1024 a^3 x^3\right ){}^3}}}+e^{3 c_1}}{a^2 x^2}}}-4608 a x-\frac {36 \sqrt [3]{2} \sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {e^{3 c_1} \left (e^{3 c_1}-1024 a^3 x^3\right ){}^3}}}{a x}-\frac {576\ 2^{2/3} \left (128 a^3 x^3+e^{3 c_1}\right )}{\sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {e^{3 c_1} \left (e^{3 c_1}-1024 a^3 x^3\right ){}^3}}}+\frac {2 e^{3 c_1}}{a^2 x^2}} x+e^{\frac {3 c_1}{2}}}{72 a x}\right \},\left \{y(x)\to \frac {-a \sqrt {\frac {-2304 a^3 x^3+36 \sqrt [3]{2} a \sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {e^{3 c_1} \left (e^{3 c_1}-1024 a^3 x^3\right ){}^3}} x+\frac {576\ 2^{2/3} a \left (128 a^4 x^4+a e^{3 c_1} x\right ) x}{\sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {e^{3 c_1} \left (e^{3 c_1}-1024 a^3 x^3\right ){}^3}}}+e^{3 c_1}}{a^2 x^2}} x+a \sqrt {-\frac {2 e^{\frac {3 c_1}{2}} \left (27648+\frac {e^{3 c_1}}{a^3 x^3}\right )}{\sqrt {\frac {-2304 a^3 x^3+36 \sqrt [3]{2} a \sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {e^{3 c_1} \left (e^{3 c_1}-1024 a^3 x^3\right ){}^3}} x+\frac {576\ 2^{2/3} a \left (128 a^4 x^4+a e^{3 c_1} x\right ) x}{\sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {e^{3 c_1} \left (e^{3 c_1}-1024 a^3 x^3\right ){}^3}}}+e^{3 c_1}}{a^2 x^2}}}-4608 a x-\frac {36 \sqrt [3]{2} \sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {e^{3 c_1} \left (e^{3 c_1}-1024 a^3 x^3\right ){}^3}}}{a x}-\frac {576\ 2^{2/3} \left (128 a^3 x^3+e^{3 c_1}\right )}{\sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {e^{3 c_1} \left (e^{3 c_1}-1024 a^3 x^3\right ){}^3}}}+\frac {2 e^{3 c_1}}{a^2 x^2}} x+e^{\frac {3 c_1}{2}}}{72 a x}\right \},\left \{y(x)\to \frac {a \sqrt {\frac {-2304 a^3 x^3+36 \sqrt [3]{2} a \sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {e^{3 c_1} \left (e^{3 c_1}-1024 a^3 x^3\right ){}^3}} x+\frac {576\ 2^{2/3} a \left (128 a^4 x^4+a e^{3 c_1} x\right ) x}{\sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {e^{3 c_1} \left (e^{3 c_1}-1024 a^3 x^3\right ){}^3}}}+e^{3 c_1}}{a^2 x^2}} x-\sqrt {2} a \sqrt {\frac {e^{\frac {3 c_1}{2}} \left (27648+\frac {e^{3 c_1}}{a^3 x^3}\right )}{\sqrt {\frac {-2304 a^3 x^3+36 \sqrt [3]{2} a \sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {e^{3 c_1} \left (e^{3 c_1}-1024 a^3 x^3\right ){}^3}} x+\frac {576\ 2^{2/3} a \left (128 a^4 x^4+a e^{3 c_1} x\right ) x}{\sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {e^{3 c_1} \left (e^{3 c_1}-1024 a^3 x^3\right ){}^3}}}+e^{3 c_1}}{a^2 x^2}}}-2304 a x-\frac {18 \sqrt [3]{2} \sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {e^{3 c_1} \left (e^{3 c_1}-1024 a^3 x^3\right ){}^3}}}{a x}-\frac {288\ 2^{2/3} \left (128 a^3 x^3+e^{3 c_1}\right )}{\sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {e^{3 c_1} \left (e^{3 c_1}-1024 a^3 x^3\right ){}^3}}}+\frac {e^{3 c_1}}{a^2 x^2}} x+e^{\frac {3 c_1}{2}}}{72 a x}\right \},\left \{y(x)\to \frac {a \sqrt {\frac {-2304 a^3 x^3+36 \sqrt [3]{2} a \sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {e^{3 c_1} \left (e^{3 c_1}-1024 a^3 x^3\right ){}^3}} x+\frac {576\ 2^{2/3} a \left (128 a^4 x^4+a e^{3 c_1} x\right ) x}{\sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {e^{3 c_1} \left (e^{3 c_1}-1024 a^3 x^3\right ){}^3}}}+e^{3 c_1}}{a^2 x^2}} x+\sqrt {2} a \sqrt {\frac {e^{\frac {3 c_1}{2}} \left (27648+\frac {e^{3 c_1}}{a^3 x^3}\right )}{\sqrt {\frac {-2304 a^3 x^3+36 \sqrt [3]{2} a \sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {e^{3 c_1} \left (e^{3 c_1}-1024 a^3 x^3\right ){}^3}} x+\frac {576\ 2^{2/3} a \left (128 a^4 x^4+a e^{3 c_1} x\right ) x}{\sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {e^{3 c_1} \left (e^{3 c_1}-1024 a^3 x^3\right ){}^3}}}+e^{3 c_1}}{a^2 x^2}}}-2304 a x-\frac {18 \sqrt [3]{2} \sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {e^{3 c_1} \left (e^{3 c_1}-1024 a^3 x^3\right ){}^3}}}{a x}-\frac {288\ 2^{2/3} \left (128 a^3 x^3+e^{3 c_1}\right )}{\sqrt [3]{-131072 a^6 x^6+2560 a^3 e^{3 c_1} x^3+e^{6 c_1}+\sqrt {e^{3 c_1} \left (e^{3 c_1}-1024 a^3 x^3\right ){}^3}}}+\frac {e^{3 c_1}}{a^2 x^2}} x+e^{\frac {3 c_1}{2}}}{72 a x}\right \}\right \}$

Maple
cpu = 0.164 (sec), leaf count = 146

$\left [-\frac {\textit {\_C1} \left (\frac {-y \left (x \right )+\sqrt {-4 a x +y \left (x \right )^{2}}}{x}\right )^{\frac {3}{2}} x^{2}}{\left (-y \left (x \right )+\sqrt {-4 a x +y \left (x \right )^{2}}\right )^{2}}+x +\frac {4 a \,x^{2}}{3 \left (-y \left (x \right )+\sqrt {-4 a x +y \left (x \right )^{2}}\right )^{2}} = 0, \frac {\left (\frac {-2 y \left (x \right )-2 \sqrt {-4 a x +y \left (x \right )^{2}}}{x}\right )^{\frac {3}{2}} x^{2} \textit {\_C1}}{\left (y \left (x \right )+\sqrt {-4 a x +y \left (x \right )^{2}}\right )^{2}}+x +\frac {4 a \,x^{2}}{3 \left (y \left (x \right )+\sqrt {-4 a x +y \left (x \right )^{2}}\right )^{2}} = 0\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> (E^((3*C[1])/2) - a*x*Sqrt[(E^(3*C[1]) - 2304*a^3*x^3 + (576*2^(2/3)*a
*x*(a*E^(3*C[1])*x + 128*a^4*x^4))/(E^(6*C[1]) + 2560*a^3*E^(3*C[1])*x^3 - 13107
2*a^6*x^6 + Sqrt[E^(3*C[1])*(E^(3*C[1]) - 1024*a^3*x^3)^3])^(1/3) + 36*2^(1/3)*a
*x*(E^(6*C[1]) + 2560*a^3*E^(3*C[1])*x^3 - 131072*a^6*x^6 + Sqrt[E^(3*C[1])*(E^(
3*C[1]) - 1024*a^3*x^3)^3])^(1/3))/(a^2*x^2)] - a*x*Sqrt[(2*E^(3*C[1]))/(a^2*x^2
) - 4608*a*x - (576*2^(2/3)*(E^(3*C[1]) + 128*a^3*x^3))/(E^(6*C[1]) + 2560*a^3*E
^(3*C[1])*x^3 - 131072*a^6*x^6 + Sqrt[E^(3*C[1])*(E^(3*C[1]) - 1024*a^3*x^3)^3])
^(1/3) - (36*2^(1/3)*(E^(6*C[1]) + 2560*a^3*E^(3*C[1])*x^3 - 131072*a^6*x^6 + Sq
rt[E^(3*C[1])*(E^(3*C[1]) - 1024*a^3*x^3)^3])^(1/3))/(a*x) - (2*E^((3*C[1])/2)*(
27648 + E^(3*C[1])/(a^3*x^3)))/Sqrt[(E^(3*C[1]) - 2304*a^3*x^3 + (576*2^(2/3)*a*
x*(a*E^(3*C[1])*x + 128*a^4*x^4))/(E^(6*C[1]) + 2560*a^3*E^(3*C[1])*x^3 - 131072
*a^6*x^6 + Sqrt[E^(3*C[1])*(E^(3*C[1]) - 1024*a^3*x^3)^3])^(1/3) + 36*2^(1/3)*a*
x*(E^(6*C[1]) + 2560*a^3*E^(3*C[1])*x^3 - 131072*a^6*x^6 + Sqrt[E^(3*C[1])*(E^(3
*C[1]) - 1024*a^3*x^3)^3])^(1/3))/(a^2*x^2)]])/(72*a*x)}, {y[x] -> (E^((3*C[1])/
2) - a*x*Sqrt[(E^(3*C[1]) - 2304*a^3*x^3 + (576*2^(2/3)*a*x*(a*E^(3*C[1])*x + 12
8*a^4*x^4))/(E^(6*C[1]) + 2560*a^3*E^(3*C[1])*x^3 - 131072*a^6*x^6 + Sqrt[E^(3*C
[1])*(E^(3*C[1]) - 1024*a^3*x^3)^3])^(1/3) + 36*2^(1/3)*a*x*(E^(6*C[1]) + 2560*a
^3*E^(3*C[1])*x^3 - 131072*a^6*x^6 + Sqrt[E^(3*C[1])*(E^(3*C[1]) - 1024*a^3*x^3)
^3])^(1/3))/(a^2*x^2)] + a*x*Sqrt[(2*E^(3*C[1]))/(a^2*x^2) - 4608*a*x - (576*2^(
2/3)*(E^(3*C[1]) + 128*a^3*x^3))/(E^(6*C[1]) + 2560*a^3*E^(3*C[1])*x^3 - 131072*
a^6*x^6 + Sqrt[E^(3*C[1])*(E^(3*C[1]) - 1024*a^3*x^3)^3])^(1/3) - (36*2^(1/3)*(E
^(6*C[1]) + 2560*a^3*E^(3*C[1])*x^3 - 131072*a^6*x^6 + Sqrt[E^(3*C[1])*(E^(3*C[1
]) - 1024*a^3*x^3)^3])^(1/3))/(a*x) - (2*E^((3*C[1])/2)*(27648 + E^(3*C[1])/(a^3
*x^3)))/Sqrt[(E^(3*C[1]) - 2304*a^3*x^3 + (576*2^(2/3)*a*x*(a*E^(3*C[1])*x + 128
*a^4*x^4))/(E^(6*C[1]) + 2560*a^3*E^(3*C[1])*x^3 - 131072*a^6*x^6 + Sqrt[E^(3*C[
1])*(E^(3*C[1]) - 1024*a^3*x^3)^3])^(1/3) + 36*2^(1/3)*a*x*(E^(6*C[1]) + 2560*a^
3*E^(3*C[1])*x^3 - 131072*a^6*x^6 + Sqrt[E^(3*C[1])*(E^(3*C[1]) - 1024*a^3*x^3)^
3])^(1/3))/(a^2*x^2)]])/(72*a*x)}, {y[x] -> (E^((3*C[1])/2) + a*x*Sqrt[(E^(3*C[1
]) - 2304*a^3*x^3 + (576*2^(2/3)*a*x*(a*E^(3*C[1])*x + 128*a^4*x^4))/(E^(6*C[1])
 + 2560*a^3*E^(3*C[1])*x^3 - 131072*a^6*x^6 + Sqrt[E^(3*C[1])*(E^(3*C[1]) - 1024
*a^3*x^3)^3])^(1/3) + 36*2^(1/3)*a*x*(E^(6*C[1]) + 2560*a^3*E^(3*C[1])*x^3 - 131
072*a^6*x^6 + Sqrt[E^(3*C[1])*(E^(3*C[1]) - 1024*a^3*x^3)^3])^(1/3))/(a^2*x^2)]
- Sqrt[2]*a*x*Sqrt[E^(3*C[1])/(a^2*x^2) - 2304*a*x - (288*2^(2/3)*(E^(3*C[1]) +
128*a^3*x^3))/(E^(6*C[1]) + 2560*a^3*E^(3*C[1])*x^3 - 131072*a^6*x^6 + Sqrt[E^(3
*C[1])*(E^(3*C[1]) - 1024*a^3*x^3)^3])^(1/3) - (18*2^(1/3)*(E^(6*C[1]) + 2560*a^
3*E^(3*C[1])*x^3 - 131072*a^6*x^6 + Sqrt[E^(3*C[1])*(E^(3*C[1]) - 1024*a^3*x^3)^
3])^(1/3))/(a*x) + (E^((3*C[1])/2)*(27648 + E^(3*C[1])/(a^3*x^3)))/Sqrt[(E^(3*C[
1]) - 2304*a^3*x^3 + (576*2^(2/3)*a*x*(a*E^(3*C[1])*x + 128*a^4*x^4))/(E^(6*C[1]
) + 2560*a^3*E^(3*C[1])*x^3 - 131072*a^6*x^6 + Sqrt[E^(3*C[1])*(E^(3*C[1]) - 102
4*a^3*x^3)^3])^(1/3) + 36*2^(1/3)*a*x*(E^(6*C[1]) + 2560*a^3*E^(3*C[1])*x^3 - 13
1072*a^6*x^6 + Sqrt[E^(3*C[1])*(E^(3*C[1]) - 1024*a^3*x^3)^3])^(1/3))/(a^2*x^2)]
])/(72*a*x)}, {y[x] -> (E^((3*C[1])/2) + a*x*Sqrt[(E^(3*C[1]) - 2304*a^3*x^3 + (
576*2^(2/3)*a*x*(a*E^(3*C[1])*x + 128*a^4*x^4))/(E^(6*C[1]) + 2560*a^3*E^(3*C[1]
)*x^3 - 131072*a^6*x^6 + Sqrt[E^(3*C[1])*(E^(3*C[1]) - 1024*a^3*x^3)^3])^(1/3) +
 36*2^(1/3)*a*x*(E^(6*C[1]) + 2560*a^3*E^(3*C[1])*x^3 - 131072*a^6*x^6 + Sqrt[E^
(3*C[1])*(E^(3*C[1]) - 1024*a^3*x^3)^3])^(1/3))/(a^2*x^2)] + Sqrt[2]*a*x*Sqrt[E^
(3*C[1])/(a^2*x^2) - 2304*a*x - (288*2^(2/3)*(E^(3*C[1]) + 128*a^3*x^3))/(E^(6*C
[1]) + 2560*a^3*E^(3*C[1])*x^3 - 131072*a^6*x^6 + Sqrt[E^(3*C[1])*(E^(3*C[1]) -
1024*a^3*x^3)^3])^(1/3) - (18*2^(1/3)*(E^(6*C[1]) + 2560*a^3*E^(3*C[1])*x^3 - 13
1072*a^6*x^6 + Sqrt[E^(3*C[1])*(E^(3*C[1]) - 1024*a^3*x^3)^3])^(1/3))/(a*x) + (E
^((3*C[1])/2)*(27648 + E^(3*C[1])/(a^3*x^3)))/Sqrt[(E^(3*C[1]) - 2304*a^3*x^3 +
(576*2^(2/3)*a*x*(a*E^(3*C[1])*x + 128*a^4*x^4))/(E^(6*C[1]) + 2560*a^3*E^(3*C[1
])*x^3 - 131072*a^6*x^6 + Sqrt[E^(3*C[1])*(E^(3*C[1]) - 1024*a^3*x^3)^3])^(1/3)
+ 36*2^(1/3)*a*x*(E^(6*C[1]) + 2560*a^3*E^(3*C[1])*x^3 - 131072*a^6*x^6 + Sqrt[E
^(3*C[1])*(E^(3*C[1]) - 1024*a^3*x^3)^3])^(1/3))/(a^2*x^2)]])/(72*a*x)}}

Maple raw input

dsolve(x*diff(y(x),x)^2+y(x)*diff(y(x),x)+a = 0, y(x))

Maple raw output

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