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 = 0.309779 (sec), leaf count = 457
\[\left \{\left \{y(x)\to \frac {\sqrt [3]{2} \left (3 c_1 x^2+\sqrt {\left (3 c_1 x^2+c_3 x+6 c_2\right ){}^2+16 x^3}+c_3 x+6 c_2\right ){}^{2/3}-2\ 2^{2/3} x}{2 \sqrt [3]{3 c_1 x^2+\sqrt {\left (3 c_1 x^2+c_3 x+6 c_2\right ){}^2+16 x^3}+c_3 x+6 c_2}}\right \},\left \{y(x)\to \frac {i \left (\sqrt {3}+i\right ) \left (3 c_1 x^2+\sqrt {9 c_1^2 x^4+2 \left (3 c_1 c_3+8\right ) x^3+\left (c_3^2+36 c_1 c_2\right ) x^2+12 c_2 c_3 x+36 c_2^2}+c_3 x+6 c_2\right ){}^{2/3}+2 \sqrt [3]{2} \left (1+i \sqrt {3}\right ) x}{2\ 2^{2/3} \sqrt [3]{3 c_1 x^2+\sqrt {\left (3 c_1 x^2+c_3 x+6 c_2\right ){}^2+16 x^3}+c_3 x+6 c_2}}\right \},\left \{y(x)\to \frac {\left (-1-i \sqrt {3}\right ) \left (3 c_1 x^2+\sqrt {9 c_1^2 x^4+2 \left (3 c_1 c_3+8\right ) x^3+\left (c_3^2+36 c_1 c_2\right ) x^2+12 c_2 c_3 x+36 c_2^2}+c_3 x+6 c_2\right ){}^{2/3}+2 \sqrt [3]{2} \left (1-i \sqrt {3}\right ) x}{2\ 2^{2/3} \sqrt [3]{3 c_1 x^2+\sqrt {\left (3 c_1 x^2+c_3 x+6 c_2\right ){}^2+16 x^3}+c_3 x+6 c_2}}\right \}\right \}\]
Maple ✓
cpu = 0.049 (sec), leaf count = 26
\[ \left \{ -{\frac {{\it \_C1}\,{x}^{2}}{2}}-{\frac { \left ( y \left ( x \right ) \right ) ^{3}}{3}}-{\it \_C2}\,x-xy \left ( x \right ) +{\it \_C3}=0 \right \} \] 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^(1/3)*(1 + I*Sqrt[3])*x + I*(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))/(2*2^(2/3)*(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])^(1/3))}, {y[x] -> (2*2^(1/3)*(1 - I*Sqrt[3])*x + (-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))/(2*2^(2/3)*(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])^(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),'implicit')
Maple raw output
-1/2*_C1*x^2-1/3*y(x)^3-_C2*x-x*y(x)+_C3 = 0