#### 4.883

ODE

ODE Classification

[_dAlembert]

Book solution method
Change of variable

Mathematica
cpu = 242.704 (sec), leaf count = 286

Maple
cpu = 0.131 (sec), leaf count = 2085

Mathematica raw input

DSolve[a2 + b2*x + c2*y[x] + (a1 + b1*x + c1*y[x])*y'[x] + (a + x)*y'[x]^2 == 0,y[x],x]

Mathematica raw output

Solve[{x == ((c2 + c1*K\$1535584)*(C[1] - Integrate[(E^(((b1*c1 - (2 + c1)*c2)*Ar
cTan[(b1 + c2 + 2*(1 + c1)*K\$1535584)/Sqrt[-b1^2 + 4*b2*(1 + c1) - 2*b1*c2 - c2^
2]])/((1 + c1)*Sqrt[-b1^2 + 4*b2*(1 + c1) - 2*b1*c2 - c2^2]))*(-(a2*c1) + a1*c2
+ a*K\$1535584*(2*c2 + c1*K\$1535584)))/((c2 + c1*K\$1535584)^2*(b2 + K\$1535584*(b1
+ c2 + K\$1535584 + c1*K\$1535584))^(c1/(2 + 2*c1))), K\$1535584]))/(E^(((b1*c1 -
(2 + c1)*c2)*ArcTan[(b1 + c2 + 2*(1 + c1)*K\$1535584)/Sqrt[-b1^2 + 4*b2*(1 + c1)
- 2*b1*c2 - c2^2]])/((1 + c1)*Sqrt[-b1^2 + 4*b2*(1 + c1) - 2*b1*c2 - c2^2]))*(b2
+ K\$1535584*(b1 + c2 + K\$1535584 + c1*K\$1535584))^((2 + c1)/(2 + 2*c1))), y[x]
== -((a2 + a1*K\$1535584 + a*K\$1535584^2 + b2*x + b1*K\$1535584*x + K\$1535584^2*x)
/(c2 + c1*K\$1535584))}, {y[x], K\$1535584}]

Maple raw input

dsolve((a+x)*diff(y(x),x)^2+(a1+b1*x+c1*y(x))*diff(y(x),x)+a2+b2*x+c2*y(x) = 0, y(x),'implicit')

Maple raw output

[x(_T) = ((c1+1)*_T^2+(b1+c2)*_T+b2)^(-c1/(2*c1+2))*exp(1/(-b1^2-2*b1*c2+(4*c1+4
)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/
2))/(c1+1)*c1*b1)*exp(1/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*
_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2))/(c1+1)*c1*c2)*exp(1/(-b1^2-2*b
1*c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*
b2-c2^2)^(1/2))/(c1+1)*b1)^2*exp(1/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan
((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2))/(c1+1)*c2)^2*(_T*c1
+c2)*(Int(-1/(_T*c1+c2)^2*(_T^2*a*c1+2*_T*a*c2+a1*c2-a2*c1)*((c1+1)*_T^2+(b1+c2)
*_T+b2)^(c1/(2*c1+2))*((c1+1)*_T^2+(b1+c2)*_T+b2)^(1/(c1+1))*exp(1/(-b1^2-2*b1*c
2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-
c2^2)^(1/2))*b1)^2/exp(1/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)
*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2))/(c1+1)*c1*b1)/((c1+1)*_T^2+(b
1+c2)*_T+b2)/exp(1/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1
+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2))/(c1+1)*b1)^2/exp(1/(-b1^2-2*b1*c2+(
4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^
2)^(1/2))/(c1+1)*c2)^2/exp(1/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c
1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2))/(c1+1)*c1*c2),_T)+_C1)/((
(c1+1)*_T^2+(b1+c2)*_T+b2)^(1/(c1+1)))/exp(1/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1
/2)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2))*b1)^2, y(
_T) = (-((c1+1)*_T^2+(b1+c2)*_T+b2)^(-c1/(2*c1+2))*exp(1/(-b1^2-2*b1*c2+(4*c1+4)
*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2
))/(c1+1)*b1)^2*exp(1/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T
+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2))/(c1+1)*c2)^2*exp(1/(-b1^2-2*b1*c
2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-
c2^2)^(1/2))/(c1+1)*c1*b1)*exp(1/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((
2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2))/(c1+1)*c1*c2)*(_T*c1+
c2)*(_T^2+_T*b1+b2)*Int(-1/(_T*c1+c2)^2*(_T^2*a*c1+2*_T*a*c2+a1*c2-a2*c1)*((c1+1
)*_T^2+(b1+c2)*_T+b2)^(c1/(2*c1+2))*((c1+1)*_T^2+(b1+c2)*_T+b2)^(1/(c1+1))*exp(1
/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c
2+(4*c1+4)*b2-c2^2)^(1/2))*b1)^2/exp(1/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)*ar
ctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2))/(c1+1)*c1*b1)/(
(c1+1)*_T^2+(b1+c2)*_T+b2)/exp(1/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((
2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2))/(c1+1)*b1)^2/exp(1/(-
b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(
4*c1+4)*b2-c2^2)^(1/2))/(c1+1)*c2)^2/exp(1/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2
)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2))/(c1+1)*c1*c
2),_T)-((c1+1)*_T^2+(b1+c2)*_T+b2)^(-c1/(2*c1+2))*exp(1/(-b1^2-2*b1*c2+(4*c1+4)*
b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)
)/(c1+1)*b1)^2*exp(1/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+
b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2))/(c1+1)*c2)^2*exp(1/(-b1^2-2*b1*c2
+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c
2^2)^(1/2))/(c1+1)*c1*c2)*_C1*(_T*c1+c2)*(_T^2+_T*b1+b2)*exp(1/(-b1^2-2*b1*c2+(4
*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2
)^(1/2))/(c1+1)*c1*b1)-((c1+1)*_T^2+(b1+c2)*_T+b2)^(1/(c1+1))*exp(1/(-b1^2-2*b1*
c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1+c2)/(-b1^2-2*b1*c2+(4*c1+4)*b2
-c2^2)^(1/2))*b1)^2*(_T^2*a+_T*a1+a2))/(_T*c1+c2)/(((c1+1)*_T^2+(b1+c2)*_T+b2)^(
1/(c1+1)))/exp(1/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2)*arctan((2*(c1+1)*_T+b1+c
2)/(-b1^2-2*b1*c2+(4*c1+4)*b2-c2^2)^(1/2))*b1)^2]