4.14.39 \(y(x) \left (5 x^2+2 y(x)^2\right ) y'(x)+x \left (x^2+5 y(x)^2\right )=0\)

ODE
\[ y(x) \left (5 x^2+2 y(x)^2\right ) y'(x)+x \left (x^2+5 y(x)^2\right )=0 \] ODE Classification

[[_homogeneous, `class A`], _exact, _rational, _dAlembert]

Book solution method
Exact equation

Mathematica
cpu = 0.0415178 (sec), leaf count = 159

\[\left \{\left \{y(x)\to -\frac {\sqrt {-\sqrt {2 e^{4 c_1}+23 x^4}-5 x^2}}{\sqrt {2}}\right \},\left \{y(x)\to \frac {\sqrt {-\sqrt {2 e^{4 c_1}+23 x^4}-5 x^2}}{\sqrt {2}}\right \},\left \{y(x)\to -\frac {\sqrt {\sqrt {2 e^{4 c_1}+23 x^4}-5 x^2}}{\sqrt {2}}\right \},\left \{y(x)\to \frac {\sqrt {\sqrt {2 e^{4 c_1}+23 x^4}-5 x^2}}{\sqrt {2}}\right \}\right \}\]

Maple
cpu = 0.015 (sec), leaf count = 37

\[ \left \{ -{\frac {1}{4}\ln \left ({\frac {{x}^{4}+10\,{x}^{2} \left (y \relax (x ) \right ) ^{2}+2\, \left (y \relax (x ) \right ) ^{4}}{{x}^{4}}} \right ) }-\ln \relax (x ) -{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[x*(x^2 + 5*y[x]^2) + y[x]*(5*x^2 + 2*y[x]^2)*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> -(Sqrt[-5*x^2 - Sqrt[2*E^(4*C[1]) + 23*x^4]]/Sqrt[2])}, {y[x] -> Sqrt[
-5*x^2 - Sqrt[2*E^(4*C[1]) + 23*x^4]]/Sqrt[2]}, {y[x] -> -(Sqrt[-5*x^2 + Sqrt[2*
E^(4*C[1]) + 23*x^4]]/Sqrt[2])}, {y[x] -> Sqrt[-5*x^2 + Sqrt[2*E^(4*C[1]) + 23*x
^4]]/Sqrt[2]}}

Maple raw input

dsolve((5*x^2+2*y(x)^2)*y(x)*diff(y(x),x)+x*(x^2+5*y(x)^2) = 0, y(x),'implicit')

Maple raw output

-1/4*ln((x^4+10*x^2*y(x)^2+2*y(x)^4)/x^4)-ln(x)-_C1 = 0