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 = 0.0887001 (sec), leaf count = 441
\[\left \{\left \{y(x)\to \frac {\sqrt [3]{2} \left (\sqrt {\left (3 a x^2+c_2 x+6 c_1\right ){}^2+16 x^3}+3 a x^2+c_2 x+6 c_1\right ){}^{2/3}-2\ 2^{2/3} x}{2 \sqrt [3]{\sqrt {\left (3 a x^2+c_2 x+6 c_1\right ){}^2+16 x^3}+3 a x^2+c_2 x+6 c_1}}\right \},\left \{y(x)\to \frac {i \left (\sqrt {3}+i\right ) \left (\sqrt {9 a^2 x^4+2 x^3 \left (3 a c_2+8\right )+x^2 \left (36 a c_1+c_2^2\right )+12 c_1 c_2 x+36 c_1^2}+3 a x^2+c_2 x+6 c_1\right ){}^{2/3}+2 \sqrt [3]{2} \left (1+i \sqrt {3}\right ) x}{2\ 2^{2/3} \sqrt [3]{\sqrt {\left (3 a x^2+c_2 x+6 c_1\right ){}^2+16 x^3}+3 a x^2+c_2 x+6 c_1}}\right \},\left \{y(x)\to \frac {\left (-1-i \sqrt {3}\right ) \left (\sqrt {9 a^2 x^4+2 x^3 \left (3 a c_2+8\right )+x^2 \left (36 a c_1+c_2^2\right )+12 c_1 c_2 x+36 c_1^2}+3 a x^2+c_2 x+6 c_1\right ){}^{2/3}+2 \sqrt [3]{2} \left (1-i \sqrt {3}\right ) x}{2\ 2^{2/3} \sqrt [3]{\sqrt {\left (3 a x^2+c_2 x+6 c_1\right ){}^2+16 x^3}+3 a x^2+c_2 x+6 c_1}}\right \}\right \}\]
Maple ✓
cpu = 0.042 (sec), leaf count = 26
\[ \left \{ {\frac {a{x}^{2}}{2}}-{\frac { \left ( y \left ( x \right ) \right ) ^{3}}{3}}-{\it \_C1}\,x-xy \left ( x \right ) +{\it \_C2}=0 \right \} \] 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^(1/3)*(1 + I*Sqrt[3])*x + I*(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))/(2*2^(2/3)*(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])^(1/3))}, {y[x] -> (2*2^(1/3)*(1 - I*Sqrt[3])*x + (-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))/(2*2^(2/3)*(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])^(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),'implicit')
Maple raw output
1/2*a*x^2-1/3*y(x)^3-_C1*x-x*y(x)+_C2 = 0