4.18.47 $x^2{y}^{\prime }(x{)}^2+x^2-y(x{)}^2=0$

ODE
$$x^2{y}^{\prime }(x{)}^2+x^2-y(x{)}^2=0$$ ODE Classification

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

Book solution method
No Missing Variables ODE, Solve for $y^{\prime }$

Mathematica ✓
cpu = 0.209988 (sec), leaf count = 147

$$\{\text{Solve}[x(-2c_1+2{\sinh }^{-1}\left(\frac{\sqrt{\frac{y(x)}{x}-1}}{\sqrt{2}}\right)-2\log (x)-1)=\frac{y(x{)}^2}{x}+\sqrt{\frac{y(x)+x}{x}}\sqrt{\frac{y(x)}{x}-1}y(x),y(x)],\text{Solve}[x(-2c_1+2{\sinh }^{-1}\left(\frac{\sqrt{\frac{y(x)}{x}-1}}{\sqrt{2}}\right)+2\log (x)+1)+\frac{y(x{)}^2}{x}=y(x)\sqrt{\frac{y(x)+x}{x}}\sqrt{\frac{y(x)}{x}-1},y(x)]\}$$

Maple ✓
cpu = 0.053 (sec), leaf count = 137

$$\{\ln \left(x\right)-\frac{y\left(x\right)}{2x}\sqrt{\frac{{\left(y\left(x\right)\right)}^2-x^2}{x^2}}+\frac{1}{2}\ln \left(\frac{1}{x}(\sqrt{\frac{{\left(y\left(x\right)\right)}^2-x^2}{x^2}}x+y\left(x\right))\right)+\frac{{\left(y\left(x\right)\right)}^2}{2x^2}-\mathit{\_}\mathit{C}\mathit{1}=0,\ln \left(x\right)+\frac{y\left(x\right)}{2x}\sqrt{\frac{{\left(y\left(x\right)\right)}^2-x^2}{x^2}}-\frac{1}{2}\ln \left(\frac{1}{x}(\sqrt{\frac{{\left(y\left(x\right)\right)}^2-x^2}{x^2}}x+y\left(x\right))\right)+\frac{{\left(y\left(x\right)\right)}^2}{2x^2}-\mathit{\_}\mathit{C}\mathit{1}=0\}$$ Mathematica raw input

DSolve[x^2 - y[x]^2 + x^2*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{Solve[x*(-1 + 2*ArcSinh[Sqrt[-1 + y[x]/x]/Sqrt[2]] - 2*C[1] - 2*Log[x]) == y[x]
^2/x + y[x]*Sqrt[(x + y[x])/x]*Sqrt[-1 + y[x]/x], y[x]], Solve[x*(1 + 2*ArcSinh[
Sqrt[-1 + y[x]/x]/Sqrt[2]] - 2*C[1] + 2*Log[x]) + y[x]^2/x == y[x]*Sqrt[(x + y[x
])/x]*Sqrt[-1 + y[x]/x], y[x]]}

Maple raw input

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

Maple raw output

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