ODE
\[ (a+x) y'(x)^2+y'(x) (\text {a1}+\text {b1} x+\text {c1} y(x))+\text {a2}+\text {b2} x+\text {c2} y(x)=0 \] ODE Classification
[_rational, _dAlembert]
Book solution method
Change of variable
Mathematica ✓
cpu = 237.848 (sec), leaf count = 298
\[\text {Solve}\left [\left \{x=(\text {c1} K[1]+\text {c2}) (K[1] (\text {c1} K[1]+K[1]+\text {b1}+\text {c2})+\text {b2})^{-\frac {\text {c1}+2}{2 \text {c1}+2}} \exp \left (-\frac {(\text {b1} \text {c1}-(\text {c1}+2) \text {c2}) \tan ^{-1}\left (\frac {2 (\text {c1}+1) K[1]+\text {b1}+\text {c2}}{\sqrt {-\text {b1}^2-2 \text {b1} \text {c2}+4 \text {b2} (\text {c1}+1)-\text {c2}^2}}\right )}{(\text {c1}+1) \sqrt {-\text {b1}^2-2 \text {b1} \text {c2}+4 \text {b2} (\text {c1}+1)-\text {c2}^2}}\right ) \left (-\int \frac {(K[1] (\text {c1} K[1]+K[1]+\text {b1}+\text {c2})+\text {b2})^{-\frac {\text {c1}}{2 \text {c1}+2}} (a K[1] (\text {c1} K[1]+2 \text {c2})+\text {a1} \text {c2}-\text {a2} \text {c1}) \exp \left (\frac {(\text {b1} \text {c1}-(\text {c1}+2) \text {c2}) \tan ^{-1}\left (\frac {2 (\text {c1}+1) K[1]+\text {b1}+\text {c2}}{\sqrt {-\text {b1}^2-2 \text {b1} \text {c2}+4 \text {b2} (\text {c1}+1)-\text {c2}^2}}\right )}{(\text {c1}+1) \sqrt {-\text {b1}^2-2 \text {b1} \text {c2}+4 \text {b2} (\text {c1}+1)-\text {c2}^2}}\right )}{(\text {c1} K[1]+\text {c2})^2} \, dK[1]+c_1\right ),y(x)=-\frac {K[1] ((a+x) K[1]+\text {a1}+\text {b1} x)+\text {a2}+\text {b2} x}{\text {c1} K[1]+\text {c2}}\right \},\{y(x),K[1]\}\right ]\]
Maple ✓
cpu = 35.188 (sec), leaf count = 9301
\[\text {Expression too large to display}\] 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 == ((C[1] - Integrate[(E^(((b1*c1 - (2 + c1)*c2)*ArcTan[(b1 + c2 + 2*(1 + c1)*K[1])/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[1]*(2*c2 + c1*K[1])))/((c2 + c1*K[1])^2*(b2 + K[1]*(b1 + c2 + K[1] + c1*K[1]))^(c1/(2 + 2*c1))), K[1]])*(c2 + c1*K[1]))/(E^(((b1*c1 - (2 + c1)*c2)*ArcTan[(b1 + c2 + 2*(1 + c1)*K[1])/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[1]*(b1 + c2 + K[1] + c1*K[1]))^((2 + c1)/(2 + 2*c1))), y[x] == -((a2 + b2*x + K[1]*(a1 + b1*x + (a + x)*K[1]))/(c2 + c1*K[1]))}, {y[x], K[1]}]
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))
Maple raw output
[(-b1*c1*x-c1^2*y(x)+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*c1+2*c2*a-a1*c1+2*x*c2)*(1/2*(a1^2-2*a2*x-2*a*c2*y(x)+2*a1*c1*y(x)-2*c2*x*y(x)+2*a1*c1^2*y(x)-(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*c1^2*y(x)+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*a*b1+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*a*c2-(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*a1*c1-(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*c1*y(x)+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*c2*x-2*a*b2*c1*x+2*a*b2*x+a1*b1*x+b1^2*c1*x^2-a*b1^2*x-a*a1*b1-b1*c2*x^2-a*a1*c2-a1*c2*x-2*b2*c1*x^2-2*a*a2*c1-2*a2*c1*x-2*a*a2+c1^2*y(x)^2+c1^3*y(x)^2-(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*a1+a1^2*c1+2*a^2*b2-3*a*c1*c2*y(x)-3*c1*c2*x*y(x)+2*b1*c1^2*x*y(x)+b1*c1*x*y(x)-(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*b1*c1*x-a*b1*c1*y(x)+2*a1*b1*c1*x-a*b1*c2*x)/(a+x)^2)^(-1/2*(c1+2)/(c1+1))/(a+x)*_C1*exp(1/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2)*arctan((-b1*c1*x-c1^2*y(x)+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*c1+b1*a+c2*a-a1*c1-c1*y(x)+x*c2+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)-a1)/(a+x)/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2))/(c1+1)*c1*b1+1/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2)*arctan((-b1*c1*x-c1^2*y(x)+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*c1+b1*a+c2*a-a1*c1-c1*y(x)+x*c2+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)-a1)/(a+x)/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2))/(c1+1)*c1*c2+2/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2)*arctan((-b1*c1*x-c1^2*y(x)+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*c1+b1*a+c2*a-a1*c1-c1*y(x)+x*c2+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)-a1)/(a+x)/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2))/(c1+1)*b1+2/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2)*arctan((-b1*c1*x-c1^2*y(x)+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*c1+b1*a+c2*a-a1*c1-c1*y(x)+x*c2+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)-a1)/(a+x)/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2))/(c1+1)*c2-2/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2)*arctan((-b1*c1*x-c1^2*y(x)+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*c1+b1*a+c2*a-a1*c1-c1*y(x)+x*c2+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)-a1)/(a+x)/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2))*b1)+x-1/2*(-b1*c1*x-c1^2*y(x)+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*c1+2*c2*a-a1*c1+2*x*c2)*Intat(-(_a^2*c1+_a^2+_a*b1+_a*c2+b2)^(-1/2/(c1+1)*c1)*(_a^2*a*c1+2*_a*a*c2+a1*c2-a2*c1)/(_a*c1+c2)^2*exp(2/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2)*arctan((2*_a*c1+2*_a+b1+c2)/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2))*b1-2/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2)*arctan((2*_a*c1+2*_a+b1+c2)/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2))/(c1+1)*c2-2/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2)*arctan((2*_a*c1+2*_a+b1+c2)/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2))/(c1+1)*b1-1/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2)*arctan((2*_a*c1+2*_a+b1+c2)/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2))/(c1+1)*c1*c2-1/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2)*arctan((2*_a*c1+2*_a+b1+c2)/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2))/(c1+1)*c1*b1),_a = 1/2/(a+x)*(-b1*x-c1*y(x)-a1+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)))*(1/2*(a1^2-2*a2*x-2*a*c2*y(x)+2*a1*c1*y(x)-2*c2*x*y(x)+2*a1*c1^2*y(x)-(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*c1^2*y(x)+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*a*b1+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*a*c2-(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*a1*c1-(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*c1*y(x)+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*c2*x-2*a*b2*c1*x+2*a*b2*x+a1*b1*x+b1^2*c1*x^2-a*b1^2*x-a*a1*b1-b1*c2*x^2-a*a1*c2-a1*c2*x-2*b2*c1*x^2-2*a*a2*c1-2*a2*c1*x-2*a*a2+c1^2*y(x)^2+c1^3*y(x)^2-(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*a1+a1^2*c1+2*a^2*b2-3*a*c1*c2*y(x)-3*c1*c2*x*y(x)+2*b1*c1^2*x*y(x)+b1*c1*x*y(x)-(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*b1*c1*x-a*b1*c1*y(x)+2*a1*b1*c1*x-a*b1*c2*x)/(a+x)^2)^(-1/2*(c1+2)/(c1+1))/(a+x)*exp(1/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2)*arctan((-b1*c1*x-c1^2*y(x)+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*c1+b1*a+c2*a-a1*c1-c1*y(x)+x*c2+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)-a1)/(a+x)/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2))/(c1+1)*c1*b1+1/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2)*arctan((-b1*c1*x-c1^2*y(x)+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*c1+b1*a+c2*a-a1*c1-c1*y(x)+x*c2+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)-a1)/(a+x)/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2))/(c1+1)*c1*c2+2/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2)*arctan((-b1*c1*x-c1^2*y(x)+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*c1+b1*a+c2*a-a1*c1-c1*y(x)+x*c2+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)-a1)/(a+x)/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2))/(c1+1)*b1+2/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2)*arctan((-b1*c1*x-c1^2*y(x)+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*c1+b1*a+c2*a-a1*c1-c1*y(x)+x*c2+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)-a1)/(a+x)/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2))/(c1+1)*c2-2/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2)*arctan((-b1*c1*x-c1^2*y(x)+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*c1+b1*a+c2*a-a1*c1-c1*y(x)+x*c2+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)-a1)/(a+x)/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2))*b1) = 0, (b1*c1*x+c1^2*y(x)+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*c1-2*c2*a+a1*c1-2*x*c2)*(-1/2*(-a1^2+2*a2*x+2*a*c2*y(x)-2*a1*c1*y(x)+2*c2*x*y(x)-2*a1*c1^2*y(x)-(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*c1^2*y(x)+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*a*b1+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*a*c2-(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*a1*c1-(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*c1*y(x)+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*c2*x+2*a*b2*c1*x-2*a*b2*x-a1*b1*x-b1^2*c1*x^2+a*b1^2*x+a*a1*b1+b1*c2*x^2+a*a1*c2+a1*c2*x+2*b2*c1*x^2+2*a*a2*c1+2*a2*c1*x+2*a*a2-c1^2*y(x)^2-c1^3*y(x)^2-(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*a1-a1^2*c1-2*a^2*b2+3*a*c1*c2*y(x)+3*c1*c2*x*y(x)-2*b1*c1^2*x*y(x)-b1*c1*x*y(x)-(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*b1*c1*x+a*b1*c1*y(x)-2*a1*b1*c1*x+a*b1*c2*x)/(a+x)^2)^(-1/2*(c1+2)/(c1+1))/(a+x)*_C1*exp(-1/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2)*arctan((b1*c1*x+c1^2*y(x)+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*c1-b1*a-c2*a+a1*c1+c1*y(x)-x*c2+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)+a1)/(a+x)/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2))/(c1+1)*c1*b1-1/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2)*arctan((b1*c1*x+c1^2*y(x)+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*c1-b1*a-c2*a+a1*c1+c1*y(x)-x*c2+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)+a1)/(a+x)/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2))/(c1+1)*c1*c2-2/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2)*arctan((b1*c1*x+c1^2*y(x)+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*c1-b1*a-c2*a+a1*c1+c1*y(x)-x*c2+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)+a1)/(a+x)/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2))/(c1+1)*b1-2/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2)*arctan((b1*c1*x+c1^2*y(x)+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*c1-b1*a-c2*a+a1*c1+c1*y(x)-x*c2+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)+a1)/(a+x)/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2))/(c1+1)*c2+2/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2)*arctan((b1*c1*x+c1^2*y(x)+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*c1-b1*a-c2*a+a1*c1+c1*y(x)-x*c2+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)+a1)/(a+x)/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2))*b1)+x+1/2*(b1*c1*x+c1^2*y(x)+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*c1-2*c2*a+a1*c1-2*x*c2)*Intat(-(_a^2*c1+_a^2+_a*b1+_a*c2+b2)^(-1/2/(c1+1)*c1)*(_a^2*a*c1+2*_a*a*c2+a1*c2-a2*c1)/(_a*c1+c2)^2*exp(2/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2)*arctan((2*_a*c1+2*_a+b1+c2)/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2))*b1-2/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2)*arctan((2*_a*c1+2*_a+b1+c2)/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2))/(c1+1)*c2-2/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2)*arctan((2*_a*c1+2*_a+b1+c2)/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2))/(c1+1)*b1-1/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2)*arctan((2*_a*c1+2*_a+b1+c2)/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2))/(c1+1)*c1*c2-1/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2)*arctan((2*_a*c1+2*_a+b1+c2)/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2))/(c1+1)*c1*b1),_a = -1/2*(b1*x+c1*y(x)+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)+a1)/(a+x))*(-1/2*(-a1^2+2*a2*x+2*a*c2*y(x)-2*a1*c1*y(x)+2*c2*x*y(x)-2*a1*c1^2*y(x)-(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*c1^2*y(x)+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*a*b1+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*a*c2-(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*a1*c1-(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*c1*y(x)+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*c2*x+2*a*b2*c1*x-2*a*b2*x-a1*b1*x-b1^2*c1*x^2+a*b1^2*x+a*a1*b1+b1*c2*x^2+a*a1*c2+a1*c2*x+2*b2*c1*x^2+2*a*a2*c1+2*a2*c1*x+2*a*a2-c1^2*y(x)^2-c1^3*y(x)^2-(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*a1-a1^2*c1-2*a^2*b2+3*a*c1*c2*y(x)+3*c1*c2*x*y(x)-2*b1*c1^2*x*y(x)-b1*c1*x*y(x)-(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*b1*c1*x+a*b1*c1*y(x)-2*a1*b1*c1*x+a*b1*c2*x)/(a+x)^2)^(-1/2*(c1+2)/(c1+1))/(a+x)*exp(-1/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2)*arctan((b1*c1*x+c1^2*y(x)+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*c1-b1*a-c2*a+a1*c1+c1*y(x)-x*c2+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)+a1)/(a+x)/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2))/(c1+1)*c1*b1-1/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2)*arctan((b1*c1*x+c1^2*y(x)+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*c1-b1*a-c2*a+a1*c1+c1*y(x)-x*c2+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)+a1)/(a+x)/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2))/(c1+1)*c1*c2-2/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2)*arctan((b1*c1*x+c1^2*y(x)+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*c1-b1*a-c2*a+a1*c1+c1*y(x)-x*c2+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)+a1)/(a+x)/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2))/(c1+1)*b1-2/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2)*arctan((b1*c1*x+c1^2*y(x)+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*c1-b1*a-c2*a+a1*c1+c1*y(x)-x*c2+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)+a1)/(a+x)/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2))/(c1+1)*c2+2/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2)*arctan((b1*c1*x+c1^2*y(x)+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)*c1-b1*a-c2*a+a1*c1+c1*y(x)-x*c2+(b1^2*x^2+2*b1*c1*x*y(x)+c1^2*y(x)^2-4*a*b2*x-4*a*c2*y(x)+2*a1*b1*x+2*a1*c1*y(x)-4*b2*x^2-4*c2*x*y(x)-4*a*a2+a1^2-4*a2*x)^(1/2)+a1)/(a+x)/(-b1^2-2*b1*c2+4*b2*c1-c2^2+4*b2)^(1/2))*b1) = 0]