##### 4.16.5 $$y'(x)^2=x^2+y(x)$$

ODE
$y'(x)^2=x^2+y(x)$ ODE Classiﬁcation

[[_homogeneous, class G]]

Book solution method
Homogeneous ODE, The Isobaric equation

Mathematica
cpu = 1.25204 (sec), leaf count = 410

$\left \{\text {Solve}\left [c_1=\int _1^{y(x)}\frac {\sqrt {x^2+K[2]} x-2 K[2]-\left (x^4+K[2] x^2-4 K[2]^2\right ) \int _1^x-\frac {\left (2 K[1]^2+K[2]\right ) \left (K[1]^4+K[2] K[1]^2-4 K[2] \sqrt {K[1]^2+K[2]} K[1]+4 K[2]^2\right )}{\sqrt {K[1]^2+K[2]} \left (K[1]^4+K[2] K[1]^2-4 K[2]^2\right )^2}dK[1]}{x^4+K[2] x^2-4 K[2]^2}dK[2]+\int _1^x\frac {K[1]^3+y(x) \left (K[1]-2 \sqrt {K[1]^2+y(x)}\right )}{K[1]^4+y(x) K[1]^2-4 y(x)^2}dK[1],y(x)\right ],\text {Solve}\left [c_1=\int _1^{y(x)}-\frac {\sqrt {x^2+K[4]} x+2 K[4]+\left (x^4+K[4] x^2-4 K[4]^2\right ) \int _1^x\frac {\left (2 K[3]^2+K[4]\right ) \left (K[3]^4+K[4] K[3]^2+4 K[4] \sqrt {K[3]^2+K[4]} K[3]+4 K[4]^2\right )}{\sqrt {K[3]^2+K[4]} \left (K[3]^4+K[4] K[3]^2-4 K[4]^2\right )^2}dK[3]}{x^4+K[4] x^2-4 K[4]^2}dK[4]+\int _1^x\frac {K[3]^3+y(x) \left (K[3]+2 \sqrt {K[3]^2+y(x)}\right )}{K[3]^4+y(x) K[3]^2-4 y(x)^2}dK[3],y(x)\right ]\right \}$

Maple
cpu = 0.369 (sec), leaf count = 281

$\left [-\ln \left (\sqrt {x^{2}+y \left (x \right )}\, x +2 y \left (x \right )\right )-\frac {2 \sqrt {17}\, \arctanh \left (\frac {\left (4 \sqrt {x^{2}+y \left (x \right )}+x \right ) \sqrt {17}}{17 x}\right )}{17}+\ln \left (-\sqrt {x^{2}+y \left (x \right )}\, x +2 y \left (x \right )\right )-\frac {2 \sqrt {17}\, \arctanh \left (\frac {\left (-x +4 \sqrt {x^{2}+y \left (x \right )}\right ) \sqrt {17}}{17 x}\right )}{17}+\ln \left (-x^{4}-x^{2} y \left (x \right )+4 y \left (x \right )^{2}\right )-\frac {2 \sqrt {17}\, \arctanh \left (\frac {\left (-x^{2}+8 y \left (x \right )\right ) \sqrt {17}}{17 x^{2}}\right )}{17}-\textit {\_C1} = 0, -\ln \left (\sqrt {x^{2}+y \left (x \right )}\, x +2 y \left (x \right )\right )-\frac {2 \sqrt {17}\, \arctanh \left (\frac {\left (4 \sqrt {x^{2}+y \left (x \right )}+x \right ) \sqrt {17}}{17 x}\right )}{17}+\ln \left (-\sqrt {x^{2}+y \left (x \right )}\, x +2 y \left (x \right )\right )-\frac {2 \sqrt {17}\, \arctanh \left (\frac {\left (-x +4 \sqrt {x^{2}+y \left (x \right )}\right ) \sqrt {17}}{17 x}\right )}{17}-\ln \left (-x^{4}-x^{2} y \left (x \right )+4 y \left (x \right )^{2}\right )+\frac {2 \sqrt {17}\, \arctanh \left (\frac {\left (-x^{2}+8 y \left (x \right )\right ) \sqrt {17}}{17 x^{2}}\right )}{17}-\textit {\_C1} = 0\right ]$ Mathematica raw input

DSolve[y'[x]^2 == x^2 + y[x],y[x],x]

Mathematica raw output

{Solve[C[1] == Inactive[Integrate][(K[1]^3 + y[x]*(K[1] - 2*Sqrt[K[1]^2 + y[x]])
)/(K[1]^4 + K[1]^2*y[x] - 4*y[x]^2), {K[1], 1, x}] + Inactive[Integrate][(-2*K[2
] + x*Sqrt[x^2 + K[2]] - (x^4 + x^2*K[2] - 4*K[2]^2)*Inactive[Integrate][-(((2*K
[1]^2 + K[2])*(K[1]^4 + K[1]^2*K[2] + 4*K[2]^2 - 4*K[1]*K[2]*Sqrt[K[1]^2 + K[2]]
))/(Sqrt[K[1]^2 + K[2]]*(K[1]^4 + K[1]^2*K[2] - 4*K[2]^2)^2)), {K[1], 1, x}])/(x
^4 + x^2*K[2] - 4*K[2]^2), {K[2], 1, y[x]}], y[x]], Solve[C[1] == Inactive[Integ
rate][(K[3]^3 + y[x]*(K[3] + 2*Sqrt[K[3]^2 + y[x]]))/(K[3]^4 + K[3]^2*y[x] - 4*y
[x]^2), {K[3], 1, x}] + Inactive[Integrate][-((2*K[4] + x*Sqrt[x^2 + K[4]] + (x^
4 + x^2*K[4] - 4*K[4]^2)*Inactive[Integrate][((2*K[3]^2 + K[4])*(K[3]^4 + K[3]^2
*K[4] + 4*K[4]^2 + 4*K[3]*K[4]*Sqrt[K[3]^2 + K[4]]))/(Sqrt[K[3]^2 + K[4]]*(K[3]^
4 + K[3]^2*K[4] - 4*K[4]^2)^2), {K[3], 1, x}])/(x^4 + x^2*K[4] - 4*K[4]^2)), {K[
4], 1, y[x]}], y[x]]}

Maple raw input

dsolve(diff(y(x),x)^2 = x^2+y(x), y(x))

Maple raw output

[-ln((x^2+y(x))^(1/2)*x+2*y(x))-2/17*17^(1/2)*arctanh(1/17*(4*(x^2+y(x))^(1/2)+x
)*17^(1/2)/x)+ln(-(x^2+y(x))^(1/2)*x+2*y(x))-2/17*17^(1/2)*arctanh(1/17*(-x+4*(x
^2+y(x))^(1/2))*17^(1/2)/x)+ln(-x^4-x^2*y(x)+4*y(x)^2)-2/17*17^(1/2)*arctanh(1/1
7*(-x^2+8*y(x))*17^(1/2)/x^2)-_C1 = 0, -ln((x^2+y(x))^(1/2)*x+2*y(x))-2/17*17^(1
/2)*arctanh(1/17*(4*(x^2+y(x))^(1/2)+x)*17^(1/2)/x)+ln(-(x^2+y(x))^(1/2)*x+2*y(x
))-2/17*17^(1/2)*arctanh(1/17*(-x+4*(x^2+y(x))^(1/2))*17^(1/2)/x)-ln(-x^4-x^2*y(
x)+4*y(x)^2)+2/17*17^(1/2)*arctanh(1/17*(-x^2+8*y(x))*17^(1/2)/x^2)-_C1 = 0]