4.14.24 \(x \left (4 x^2 y(x)-12 x y(x)^2+5 x+3\right ) y'(x)+y(x) \left (6 x^2 y(x)-8 x y(x)^2+10 x+3\right )=0\)

ODE
\[ x \left (4 x^2 y(x)-12 x y(x)^2+5 x+3\right ) y'(x)+y(x) \left (6 x^2 y(x)-8 x y(x)^2+10 x+3\right )=0 \] ODE Classification

[_exact, _rational]

Book solution method
Exact equation

Mathematica
cpu = 0.0343729 (sec), leaf count = 660

\[\left \{\left \{y(x)\to -\frac {\sqrt [3]{432 c_1 x^4+8 \sqrt {\left (-54 c_1 x^4+2 x^9+45 x^7+27 x^6\right ){}^2-4 x^9 \left (x^3+15 x+9\right )^3}-16 x^9-360 x^7-216 x^6}}{12 \sqrt [3]{2} x^2}-\frac {\left (x^3+15 x+9\right ) x}{3\ 2^{2/3} \sqrt [3]{54 c_1 x^4+3 \sqrt {3} \sqrt {-x^8 \left (\left (8 c_1+30\right ) x^5+180 \left (c_1+5\right ) x^3+108 \left (c_1+5\right ) x^2-108 c_1^2+25 x^6+509 x^4+108 x\right )}-2 x^9-45 x^7-27 x^6}}+\frac {x}{6}\right \},\left \{y(x)\to \frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{432 c_1 x^4+8 \sqrt {\left (-54 c_1 x^4+2 x^9+45 x^7+27 x^6\right ){}^2-4 x^9 \left (x^3+15 x+9\right )^3}-16 x^9-360 x^7-216 x^6}}{24 \sqrt [3]{2} x^2}+\frac {\left (1+i \sqrt {3}\right ) \left (x^3+15 x+9\right ) x}{6\ 2^{2/3} \sqrt [3]{54 c_1 x^4+3 \sqrt {3} \sqrt {-x^8 \left (\left (8 c_1+30\right ) x^5+180 \left (c_1+5\right ) x^3+108 \left (c_1+5\right ) x^2-108 c_1^2+25 x^6+509 x^4+108 x\right )}-2 x^9-45 x^7-27 x^6}}+\frac {x}{6}\right \},\left \{y(x)\to \frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{432 c_1 x^4+8 \sqrt {\left (-54 c_1 x^4+2 x^9+45 x^7+27 x^6\right ){}^2-4 x^9 \left (x^3+15 x+9\right )^3}-16 x^9-360 x^7-216 x^6}}{24 \sqrt [3]{2} x^2}+\frac {\left (1-i \sqrt {3}\right ) \left (x^3+15 x+9\right ) x}{6\ 2^{2/3} \sqrt [3]{54 c_1 x^4+3 \sqrt {3} \sqrt {-x^8 \left (\left (8 c_1+30\right ) x^5+180 \left (c_1+5\right ) x^3+108 \left (c_1+5\right ) x^2-108 c_1^2+25 x^6+509 x^4+108 x\right )}-2 x^9-45 x^7-27 x^6}}+\frac {x}{6}\right \}\right \}\]

Maple
cpu = 0.024 (sec), leaf count = 35

\[ \left \{ -4\,{x}^{2} \left (y \relax (x ) \right ) ^{3}+2\,{x}^{3} \left (y \relax (x ) \right ) ^{2}+ \left (5\,{x}^{2}+3\,x \right ) y \relax (x ) +{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[y[x]*(3 + 10*x + 6*x^2*y[x] - 8*x*y[x]^2) + x*(3 + 5*x + 4*x^2*y[x] - 12*x*y[x]^2)*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> x/6 - (x*(9 + 15*x + x^3))/(3*2^(2/3)*(-27*x^6 - 45*x^7 - 2*x^9 + 54*x
^4*C[1] + 3*Sqrt[3]*Sqrt[-(x^8*(108*x + 509*x^4 + 25*x^6 - 108*C[1]^2 + 108*x^2*
(5 + C[1]) + 180*x^3*(5 + C[1]) + x^5*(30 + 8*C[1])))])^(1/3)) - (-216*x^6 - 360
*x^7 - 16*x^9 + 432*x^4*C[1] + 8*Sqrt[-4*x^9*(9 + 15*x + x^3)^3 + (27*x^6 + 45*x
^7 + 2*x^9 - 54*x^4*C[1])^2])^(1/3)/(12*2^(1/3)*x^2)}, {y[x] -> x/6 + ((1 + I*Sq
rt[3])*x*(9 + 15*x + x^3))/(6*2^(2/3)*(-27*x^6 - 45*x^7 - 2*x^9 + 54*x^4*C[1] + 
3*Sqrt[3]*Sqrt[-(x^8*(108*x + 509*x^4 + 25*x^6 - 108*C[1]^2 + 108*x^2*(5 + C[1])
 + 180*x^3*(5 + C[1]) + x^5*(30 + 8*C[1])))])^(1/3)) + ((1 - I*Sqrt[3])*(-216*x^
6 - 360*x^7 - 16*x^9 + 432*x^4*C[1] + 8*Sqrt[-4*x^9*(9 + 15*x + x^3)^3 + (27*x^6
 + 45*x^7 + 2*x^9 - 54*x^4*C[1])^2])^(1/3))/(24*2^(1/3)*x^2)}, {y[x] -> x/6 + ((
1 - I*Sqrt[3])*x*(9 + 15*x + x^3))/(6*2^(2/3)*(-27*x^6 - 45*x^7 - 2*x^9 + 54*x^4
*C[1] + 3*Sqrt[3]*Sqrt[-(x^8*(108*x + 509*x^4 + 25*x^6 - 108*C[1]^2 + 108*x^2*(5
 + C[1]) + 180*x^3*(5 + C[1]) + x^5*(30 + 8*C[1])))])^(1/3)) + ((1 + I*Sqrt[3])*
(-216*x^6 - 360*x^7 - 16*x^9 + 432*x^4*C[1] + 8*Sqrt[-4*x^9*(9 + 15*x + x^3)^3 +
 (27*x^6 + 45*x^7 + 2*x^9 - 54*x^4*C[1])^2])^(1/3))/(24*2^(1/3)*x^2)}}

Maple raw input

dsolve(x*(3+5*x-12*x*y(x)^2+4*x^2*y(x))*diff(y(x),x)+(3+10*x-8*x*y(x)^2+6*x^2*y(x))*y(x) = 0, y(x),'implicit')

Maple raw output

-4*x^2*y(x)^3+2*x^3*y(x)^2+(5*x^2+3*x)*y(x)+_C1 = 0