ODE
\[ \left (y(x)^2+x\right ) y''(x)+2 y(x) y'(x)^2+2 y'(x)=a \] ODE Classification
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.221562 (sec), leaf count = 441
\[\left \{\left \{y(x)\to \frac {-2\ 2^{2/3} x+\sqrt [3]{2} \left (3 a x^2+\sqrt {16 x^3+\left (3 a x^2+c_2 x+6 c_1\right ){}^2}+c_2 x+6 c_1\right ){}^{2/3}}{2 \sqrt [3]{3 a x^2+\sqrt {16 x^3+\left (3 a x^2+c_2 x+6 c_1\right ){}^2}+c_2 x+6 c_1}}\right \},\left \{y(x)\to \frac {i \sqrt [3]{2} \left (\sqrt {3}+i\right ) \left (\sqrt {9 a^2 x^4+2 x^3 (8+3 a c_2)+x^2 \left (36 a c_1+c_2{}^2\right )+12 c_1 c_2 x+36 c_1{}^2}+3 a x^2+c_2 x+6 c_1\right ){}^{2/3}+2\ 2^{2/3} \left (1+i \sqrt {3}\right ) x}{4 \sqrt [3]{3 a x^2+\sqrt {16 x^3+\left (3 a x^2+c_2 x+6 c_1\right ){}^2}+c_2 x+6 c_1}}\right \},\left \{y(x)\to \frac {\sqrt [3]{2} \left (-1-i \sqrt {3}\right ) \left (\sqrt {9 a^2 x^4+2 x^3 (8+3 a c_2)+x^2 \left (36 a c_1+c_2{}^2\right )+12 c_1 c_2 x+36 c_1{}^2}+3 a x^2+c_2 x+6 c_1\right ){}^{2/3}+2\ 2^{2/3} \left (1-i \sqrt {3}\right ) x}{4 \sqrt [3]{3 a x^2+\sqrt {16 x^3+\left (3 a x^2+c_2 x+6 c_1\right ){}^2}+c_2 x+6 c_1}}\right \}\right \}\]
Maple ✓
cpu = 0.235 (sec), leaf count = 710
\[\left [y \left (x \right ) = \frac {\left (6 a \,x^{2}-12 \textit {\_C1} x +12 \textit {\_C2} +2 \sqrt {9 a^{2} x^{4}-36 a \,x^{3} \textit {\_C1} +36 \textit {\_C1}^{2} x^{2}+36 a \,x^{2} \textit {\_C2} -72 \textit {\_C1} x \textit {\_C2} +16 x^{3}+36 \textit {\_C2}^{2}}\right )^{\frac {1}{3}}}{2}-\frac {2 x}{\left (6 a \,x^{2}-12 \textit {\_C1} x +12 \textit {\_C2} +2 \sqrt {9 a^{2} x^{4}-36 a \,x^{3} \textit {\_C1} +36 \textit {\_C1}^{2} x^{2}+36 a \,x^{2} \textit {\_C2} -72 \textit {\_C1} x \textit {\_C2} +16 x^{3}+36 \textit {\_C2}^{2}}\right )^{\frac {1}{3}}}, y \left (x \right ) = -\frac {\left (6 a \,x^{2}-12 \textit {\_C1} x +12 \textit {\_C2} +2 \sqrt {9 a^{2} x^{4}-36 a \,x^{3} \textit {\_C1} +36 \textit {\_C1}^{2} x^{2}+36 a \,x^{2} \textit {\_C2} -72 \textit {\_C1} x \textit {\_C2} +16 x^{3}+36 \textit {\_C2}^{2}}\right )^{\frac {1}{3}}}{4}+\frac {x}{\left (6 a \,x^{2}-12 \textit {\_C1} x +12 \textit {\_C2} +2 \sqrt {9 a^{2} x^{4}-36 a \,x^{3} \textit {\_C1} +36 \textit {\_C1}^{2} x^{2}+36 a \,x^{2} \textit {\_C2} -72 \textit {\_C1} x \textit {\_C2} +16 x^{3}+36 \textit {\_C2}^{2}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (6 a \,x^{2}-12 \textit {\_C1} x +12 \textit {\_C2} +2 \sqrt {9 a^{2} x^{4}-36 a \,x^{3} \textit {\_C1} +36 \textit {\_C1}^{2} x^{2}+36 a \,x^{2} \textit {\_C2} -72 \textit {\_C1} x \textit {\_C2} +16 x^{3}+36 \textit {\_C2}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {2 x}{\left (6 a \,x^{2}-12 \textit {\_C1} x +12 \textit {\_C2} +2 \sqrt {9 a^{2} x^{4}-36 a \,x^{3} \textit {\_C1} +36 \textit {\_C1}^{2} x^{2}+36 a \,x^{2} \textit {\_C2} -72 \textit {\_C1} x \textit {\_C2} +16 x^{3}+36 \textit {\_C2}^{2}}\right )^{\frac {1}{3}}}\right )}{2}, y \left (x \right ) = -\frac {\left (6 a \,x^{2}-12 \textit {\_C1} x +12 \textit {\_C2} +2 \sqrt {9 a^{2} x^{4}-36 a \,x^{3} \textit {\_C1} +36 \textit {\_C1}^{2} x^{2}+36 a \,x^{2} \textit {\_C2} -72 \textit {\_C1} x \textit {\_C2} +16 x^{3}+36 \textit {\_C2}^{2}}\right )^{\frac {1}{3}}}{4}+\frac {x}{\left (6 a \,x^{2}-12 \textit {\_C1} x +12 \textit {\_C2} +2 \sqrt {9 a^{2} x^{4}-36 a \,x^{3} \textit {\_C1} +36 \textit {\_C1}^{2} x^{2}+36 a \,x^{2} \textit {\_C2} -72 \textit {\_C1} x \textit {\_C2} +16 x^{3}+36 \textit {\_C2}^{2}}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (6 a \,x^{2}-12 \textit {\_C1} x +12 \textit {\_C2} +2 \sqrt {9 a^{2} x^{4}-36 a \,x^{3} \textit {\_C1} +36 \textit {\_C1}^{2} x^{2}+36 a \,x^{2} \textit {\_C2} -72 \textit {\_C1} x \textit {\_C2} +16 x^{3}+36 \textit {\_C2}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {2 x}{\left (6 a \,x^{2}-12 \textit {\_C1} x +12 \textit {\_C2} +2 \sqrt {9 a^{2} x^{4}-36 a \,x^{3} \textit {\_C1} +36 \textit {\_C1}^{2} x^{2}+36 a \,x^{2} \textit {\_C2} -72 \textit {\_C1} x \textit {\_C2} +16 x^{3}+36 \textit {\_C2}^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right ]\] Mathematica raw input
DSolve[2*y'[x] + 2*y[x]*y'[x]^2 + (x + y[x]^2)*y''[x] == a,y[x],x]
Mathematica raw output
{{y[x] -> (-2*2^(2/3)*x + 2^(1/3)*(3*a*x^2 + 6*C[1] + x*C[2] + Sqrt[16*x^3 + (3*a*x^2 + 6*C[1] + x*C[2])^2])^(2/3))/(2*(3*a*x^2 + 6*C[1] + x*C[2] + Sqrt[16*x^3 + (3*a*x^2 + 6*C[1] + x*C[2])^2])^(1/3))}, {y[x] -> (2*2^(2/3)*(1 + I*Sqrt[3])*x + I*2^(1/3)*(I + Sqrt[3])*(3*a*x^2 + 6*C[1] + x*C[2] + Sqrt[9*a^2*x^4 + 36*C[1]^2 + 12*x*C[1]*C[2] + 2*x^3*(8 + 3*a*C[2]) + x^2*(36*a*C[1] + C[2]^2)])^(2/3))/(4*(3*a*x^2 + 6*C[1] + x*C[2] + Sqrt[16*x^3 + (3*a*x^2 + 6*C[1] + x*C[2])^2])^(1/3))}, {y[x] -> (2*2^(2/3)*(1 - I*Sqrt[3])*x + 2^(1/3)*(-1 - I*Sqrt[3])*(3*a*x^2 + 6*C[1] + x*C[2] + Sqrt[9*a^2*x^4 + 36*C[1]^2 + 12*x*C[1]*C[2] + 2*x^3*(8 + 3*a*C[2]) + x^2*(36*a*C[1] + C[2]^2)])^(2/3))/(4*(3*a*x^2 + 6*C[1] + x*C[2] + Sqrt[16*x^3 + (3*a*x^2 + 6*C[1] + x*C[2])^2])^(1/3))}}
Maple raw input
dsolve((x+y(x)^2)*diff(diff(y(x),x),x)+2*y(x)*diff(y(x),x)^2+2*diff(y(x),x) = a, y(x))
Maple raw output
[y(x) = 1/2*(6*a*x^2-12*_C1*x+12*_C2+2*(9*a^2*x^4-36*_C1*a*x^3+36*_C1^2*x^2+36*_C2*a*x^2-72*_C1*_C2*x+16*x^3+36*_C2^2)^(1/2))^(1/3)-2*x/(6*a*x^2-12*_C1*x+12*_C2+2*(9*a^2*x^4-36*_C1*a*x^3+36*_C1^2*x^2+36*_C2*a*x^2-72*_C1*_C2*x+16*x^3+36*_C2^2)^(1/2))^(1/3), y(x) = -1/4*(6*a*x^2-12*_C1*x+12*_C2+2*(9*a^2*x^4-36*_C1*a*x^3+36*_C1^2*x^2+36*_C2*a*x^2-72*_C1*_C2*x+16*x^3+36*_C2^2)^(1/2))^(1/3)+x/(6*a*x^2-12*_C1*x+12*_C2+2*(9*a^2*x^4-36*_C1*a*x^3+36*_C1^2*x^2+36*_C2*a*x^2-72*_C1*_C2*x+16*x^3+36*_C2^2)^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/2*(6*a*x^2-12*_C1*x+12*_C2+2*(9*a^2*x^4-36*_C1*a*x^3+36*_C1^2*x^2+36*_C2*a*x^2-72*_C1*_C2*x+16*x^3+36*_C2^2)^(1/2))^(1/3)+2*x/(6*a*x^2-12*_C1*x+12*_C2+2*(9*a^2*x^4-36*_C1*a*x^3+36*_C1^2*x^2+36*_C2*a*x^2-72*_C1*_C2*x+16*x^3+36*_C2^2)^(1/2))^(1/3)), y(x) = -1/4*(6*a*x^2-12*_C1*x+12*_C2+2*(9*a^2*x^4-36*_C1*a*x^3+36*_C1^2*x^2+36*_C2*a*x^2-72*_C1*_C2*x+16*x^3+36*_C2^2)^(1/2))^(1/3)+x/(6*a*x^2-12*_C1*x+12*_C2+2*(9*a^2*x^4-36*_C1*a*x^3+36*_C1^2*x^2+36*_C2*a*x^2-72*_C1*_C2*x+16*x^3+36*_C2^2)^(1/2))^(1/3)+1/2*I*3^(1/2)*(1/2*(6*a*x^2-12*_C1*x+12*_C2+2*(9*a^2*x^4-36*_C1*a*x^3+36*_C1^2*x^2+36*_C2*a*x^2-72*_C1*_C2*x+16*x^3+36*_C2^2)^(1/2))^(1/3)+2*x/(6*a*x^2-12*_C1*x+12*_C2+2*(9*a^2*x^4-36*_C1*a*x^3+36*_C1^2*x^2+36*_C2*a*x^2-72*_C1*_C2*x+16*x^3+36*_C2^2)^(1/2))^(1/3))]