##### 4.3.2 $$x^3+y'(x)=x \sqrt {x^4+4 y(x)}$$

ODE
$x^3+y'(x)=x \sqrt {x^4+4 y(x)}$ ODE Classiﬁcation

[[_1st_order, _with_linear_symmetries]]

Book solution method
Homogeneous equation, isobaric equation

Mathematica
cpu = 0.32659 (sec), leaf count = 63

$\left \{\left \{y(x)\to (\cosh (2 c_1)+\sinh (2 c_1)) \left (-x^2+\cosh (2 c_1)+\sinh (2 c_1)\right )\right \},\left \{y(x)\to (\cosh (2 c_1)+\sinh (2 c_1)) \left (x^2+\cosh (2 c_1)+\sinh (2 c_1)\right )\right \}\right \}$

Maple
cpu = 0.068 (sec), leaf count = 59

$\left [-\frac {y \left (x \right ) x^{2}}{x^{2}+\sqrt {x^{4}+4 y \left (x \right )}}+\frac {y \left (x \right ) \sqrt {x^{4}+4 y \left (x \right )}}{x^{2}+\sqrt {x^{4}+4 y \left (x \right )}}-\textit {\_C1} = 0\right ]$ Mathematica raw input

DSolve[x^3 + y'[x] == x*Sqrt[x^4 + 4*y[x]],y[x],x]

Mathematica raw output

{{y[x] -> (Cosh[2*C[1]] + Sinh[2*C[1]])*(-x^2 + Cosh[2*C[1]] + Sinh[2*C[1]])}, {
y[x] -> (Cosh[2*C[1]] + Sinh[2*C[1]])*(x^2 + Cosh[2*C[1]] + Sinh[2*C[1]])}}

Maple raw input

dsolve(diff(y(x),x)+x^3 = x*(x^4+4*y(x))^(1/2), y(x))

Maple raw output

[-y(x)/(x^2+(x^4+4*y(x))^(1/2))*x^2+y(x)/(x^2+(x^4+4*y(x))^(1/2))*(x^4+4*y(x))^(
1/2)-_C1 = 0]