ODE
\[ \left (y(x)^2+x\right ) y'''(x)+3 y''(x)+2 y'(x)^3+6 y(x) y'(x) y''(x)=0 \] ODE Classification
[[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.541638 (sec), leaf count = 457
\[\left \{\left \{y(x)\to \frac {-2\ 2^{2/3} x+\sqrt [3]{2} \left (3 c_1 x^2+\sqrt {16 x^3+\left (3 c_1 x^2+c_3 x+6 c_2\right ){}^2}+c_3 x+6 c_2\right ){}^{2/3}}{2 \sqrt [3]{3 c_1 x^2+\sqrt {16 x^3+\left (3 c_1 x^2+c_3 x+6 c_2\right ){}^2}+c_3 x+6 c_2}}\right \},\left \{y(x)\to \frac {i \sqrt [3]{2} \left (\sqrt {3}+i\right ) \left (3 c_1 x^2+\sqrt {9 c_1{}^2 x^4+2 (8+3 c_1 c_3) x^3+\left (c_3{}^2+36 c_1 c_2\right ) x^2+12 c_2 c_3 x+36 c_2{}^2}+c_3 x+6 c_2\right ){}^{2/3}+2\ 2^{2/3} \left (1+i \sqrt {3}\right ) x}{4 \sqrt [3]{3 c_1 x^2+\sqrt {16 x^3+\left (3 c_1 x^2+c_3 x+6 c_2\right ){}^2}+c_3 x+6 c_2}}\right \},\left \{y(x)\to \frac {\sqrt [3]{2} \left (-1-i \sqrt {3}\right ) \left (3 c_1 x^2+\sqrt {9 c_1{}^2 x^4+2 (8+3 c_1 c_3) x^3+\left (c_3{}^2+36 c_1 c_2\right ) x^2+12 c_2 c_3 x+36 c_2{}^2}+c_3 x+6 c_2\right ){}^{2/3}+2\ 2^{2/3} \left (1-i \sqrt {3}\right ) x}{4 \sqrt [3]{3 c_1 x^2+\sqrt {16 x^3+\left (3 c_1 x^2+c_3 x+6 c_2\right ){}^2}+c_3 x+6 c_2}}\right \}\right \}\]
Maple ✓
cpu = 0.525 (sec), leaf count = 710
\[\left [y \left (x \right ) = \frac {\left (-6 x^{2} \textit {\_C1} -12 \textit {\_C2} x +12 \textit {\_C3} +2 \sqrt {9 \textit {\_C1}^{2} x^{4}+36 \textit {\_C1} \textit {\_C2} \,x^{3}-36 \textit {\_C1} \textit {\_C3} \,x^{2}+36 \textit {\_C2}^{2} x^{2}-72 x \textit {\_C2} \textit {\_C3} +16 x^{3}+36 \textit {\_C3}^{2}}\right )^{\frac {1}{3}}}{2}-\frac {2 x}{\left (-6 x^{2} \textit {\_C1} -12 \textit {\_C2} x +12 \textit {\_C3} +2 \sqrt {9 \textit {\_C1}^{2} x^{4}+36 \textit {\_C1} \textit {\_C2} \,x^{3}-36 \textit {\_C1} \textit {\_C3} \,x^{2}+36 \textit {\_C2}^{2} x^{2}-72 x \textit {\_C2} \textit {\_C3} +16 x^{3}+36 \textit {\_C3}^{2}}\right )^{\frac {1}{3}}}, y \left (x \right ) = -\frac {\left (-6 x^{2} \textit {\_C1} -12 \textit {\_C2} x +12 \textit {\_C3} +2 \sqrt {9 \textit {\_C1}^{2} x^{4}+36 \textit {\_C1} \textit {\_C2} \,x^{3}-36 \textit {\_C1} \textit {\_C3} \,x^{2}+36 \textit {\_C2}^{2} x^{2}-72 x \textit {\_C2} \textit {\_C3} +16 x^{3}+36 \textit {\_C3}^{2}}\right )^{\frac {1}{3}}}{4}+\frac {x}{\left (-6 x^{2} \textit {\_C1} -12 \textit {\_C2} x +12 \textit {\_C3} +2 \sqrt {9 \textit {\_C1}^{2} x^{4}+36 \textit {\_C1} \textit {\_C2} \,x^{3}-36 \textit {\_C1} \textit {\_C3} \,x^{2}+36 \textit {\_C2}^{2} x^{2}-72 x \textit {\_C2} \textit {\_C3} +16 x^{3}+36 \textit {\_C3}^{2}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (-6 x^{2} \textit {\_C1} -12 \textit {\_C2} x +12 \textit {\_C3} +2 \sqrt {9 \textit {\_C1}^{2} x^{4}+36 \textit {\_C1} \textit {\_C2} \,x^{3}-36 \textit {\_C1} \textit {\_C3} \,x^{2}+36 \textit {\_C2}^{2} x^{2}-72 x \textit {\_C2} \textit {\_C3} +16 x^{3}+36 \textit {\_C3}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {2 x}{\left (-6 x^{2} \textit {\_C1} -12 \textit {\_C2} x +12 \textit {\_C3} +2 \sqrt {9 \textit {\_C1}^{2} x^{4}+36 \textit {\_C1} \textit {\_C2} \,x^{3}-36 \textit {\_C1} \textit {\_C3} \,x^{2}+36 \textit {\_C2}^{2} x^{2}-72 x \textit {\_C2} \textit {\_C3} +16 x^{3}+36 \textit {\_C3}^{2}}\right )^{\frac {1}{3}}}\right )}{2}, y \left (x \right ) = -\frac {\left (-6 x^{2} \textit {\_C1} -12 \textit {\_C2} x +12 \textit {\_C3} +2 \sqrt {9 \textit {\_C1}^{2} x^{4}+36 \textit {\_C1} \textit {\_C2} \,x^{3}-36 \textit {\_C1} \textit {\_C3} \,x^{2}+36 \textit {\_C2}^{2} x^{2}-72 x \textit {\_C2} \textit {\_C3} +16 x^{3}+36 \textit {\_C3}^{2}}\right )^{\frac {1}{3}}}{4}+\frac {x}{\left (-6 x^{2} \textit {\_C1} -12 \textit {\_C2} x +12 \textit {\_C3} +2 \sqrt {9 \textit {\_C1}^{2} x^{4}+36 \textit {\_C1} \textit {\_C2} \,x^{3}-36 \textit {\_C1} \textit {\_C3} \,x^{2}+36 \textit {\_C2}^{2} x^{2}-72 x \textit {\_C2} \textit {\_C3} +16 x^{3}+36 \textit {\_C3}^{2}}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (-6 x^{2} \textit {\_C1} -12 \textit {\_C2} x +12 \textit {\_C3} +2 \sqrt {9 \textit {\_C1}^{2} x^{4}+36 \textit {\_C1} \textit {\_C2} \,x^{3}-36 \textit {\_C1} \textit {\_C3} \,x^{2}+36 \textit {\_C2}^{2} x^{2}-72 x \textit {\_C2} \textit {\_C3} +16 x^{3}+36 \textit {\_C3}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {2 x}{\left (-6 x^{2} \textit {\_C1} -12 \textit {\_C2} x +12 \textit {\_C3} +2 \sqrt {9 \textit {\_C1}^{2} x^{4}+36 \textit {\_C1} \textit {\_C2} \,x^{3}-36 \textit {\_C1} \textit {\_C3} \,x^{2}+36 \textit {\_C2}^{2} x^{2}-72 x \textit {\_C2} \textit {\_C3} +16 x^{3}+36 \textit {\_C3}^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right ]\] Mathematica raw input
DSolve[2*y'[x]^3 + 3*y''[x] + 6*y[x]*y'[x]*y''[x] + (x + y[x]^2)*y'''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (-2*2^(2/3)*x + 2^(1/3)*(3*x^2*C[1] + 6*C[2] + x*C[3] + Sqrt[16*x^3 + (3*x^2*C[1] + 6*C[2] + x*C[3])^2])^(2/3))/(2*(3*x^2*C[1] + 6*C[2] + x*C[3] + Sqrt[16*x^3 + (3*x^2*C[1] + 6*C[2] + x*C[3])^2])^(1/3))}, {y[x] -> (2*2^(2/3)*(1 + I*Sqrt[3])*x + I*2^(1/3)*(I + Sqrt[3])*(3*x^2*C[1] + 6*C[2] + x*C[3] + Sqrt[9*x^4*C[1]^2 + 36*C[2]^2 + 12*x*C[2]*C[3] + 2*x^3*(8 + 3*C[1]*C[3]) + x^2*(36*C[1]*C[2] + C[3]^2)])^(2/3))/(4*(3*x^2*C[1] + 6*C[2] + x*C[3] + Sqrt[16*x^3 + (3*x^2*C[1] + 6*C[2] + x*C[3])^2])^(1/3))}, {y[x] -> (2*2^(2/3)*(1 - I*Sqrt[3])*x + 2^(1/3)*(-1 - I*Sqrt[3])*(3*x^2*C[1] + 6*C[2] + x*C[3] + Sqrt[9*x^4*C[1]^2 + 36*C[2]^2 + 12*x*C[2]*C[3] + 2*x^3*(8 + 3*C[1]*C[3]) + x^2*(36*C[1]*C[2] + C[3]^2)])^(2/3))/(4*(3*x^2*C[1] + 6*C[2] + x*C[3] + Sqrt[16*x^3 + (3*x^2*C[1] + 6*C[2] + x*C[3])^2])^(1/3))}}
Maple raw input
dsolve((x+y(x)^2)*diff(diff(diff(y(x),x),x),x)+6*y(x)*diff(y(x),x)*diff(diff(y(x),x),x)+3*diff(diff(y(x),x),x)+2*diff(y(x),x)^3 = 0, y(x))
Maple raw output
[y(x) = 1/2*(-6*x^2*_C1-12*_C2*x+12*_C3+2*(9*_C1^2*x^4+36*_C1*_C2*x^3-36*_C1*_C3*x^2+36*_C2^2*x^2-72*_C2*_C3*x+16*x^3+36*_C3^2)^(1/2))^(1/3)-2*x/(-6*x^2*_C1-12*_C2*x+12*_C3+2*(9*_C1^2*x^4+36*_C1*_C2*x^3-36*_C1*_C3*x^2+36*_C2^2*x^2-72*_C2*_C3*x+16*x^3+36*_C3^2)^(1/2))^(1/3), y(x) = -1/4*(-6*x^2*_C1-12*_C2*x+12*_C3+2*(9*_C1^2*x^4+36*_C1*_C2*x^3-36*_C1*_C3*x^2+36*_C2^2*x^2-72*_C2*_C3*x+16*x^3+36*_C3^2)^(1/2))^(1/3)+x/(-6*x^2*_C1-12*_C2*x+12*_C3+2*(9*_C1^2*x^4+36*_C1*_C2*x^3-36*_C1*_C3*x^2+36*_C2^2*x^2-72*_C2*_C3*x+16*x^3+36*_C3^2)^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/2*(-6*x^2*_C1-12*_C2*x+12*_C3+2*(9*_C1^2*x^4+36*_C1*_C2*x^3-36*_C1*_C3*x^2+36*_C2^2*x^2-72*_C2*_C3*x+16*x^3+36*_C3^2)^(1/2))^(1/3)+2*x/(-6*x^2*_C1-12*_C2*x+12*_C3+2*(9*_C1^2*x^4+36*_C1*_C2*x^3-36*_C1*_C3*x^2+36*_C2^2*x^2-72*_C2*_C3*x+16*x^3+36*_C3^2)^(1/2))^(1/3)), y(x) = -1/4*(-6*x^2*_C1-12*_C2*x+12*_C3+2*(9*_C1^2*x^4+36*_C1*_C2*x^3-36*_C1*_C3*x^2+36*_C2^2*x^2-72*_C2*_C3*x+16*x^3+36*_C3^2)^(1/2))^(1/3)+x/(-6*x^2*_C1-12*_C2*x+12*_C3+2*(9*_C1^2*x^4+36*_C1*_C2*x^3-36*_C1*_C3*x^2+36*_C2^2*x^2-72*_C2*_C3*x+16*x^3+36*_C3^2)^(1/2))^(1/3)+1/2*I*3^(1/2)*(1/2*(-6*x^2*_C1-12*_C2*x+12*_C3+2*(9*_C1^2*x^4+36*_C1*_C2*x^3-36*_C1*_C3*x^2+36*_C2^2*x^2-72*_C2*_C3*x+16*x^3+36*_C3^2)^(1/2))^(1/3)+2*x/(-6*x^2*_C1-12*_C2*x+12*_C3+2*(9*_C1^2*x^4+36*_C1*_C2*x^3-36*_C1*_C3*x^2+36*_C2^2*x^2-72*_C2*_C3*x+16*x^3+36*_C3^2)^(1/2))^(1/3))]