2.453   ODE No. 453

1. Problem in Latex
2. Mathematica input
3. Maple input

$$(a^2-1)x^2{y}^{\prime }(x{)}^2+a^2{x}^2+2xy(x)y^{\prime }(x)-y(x{)}^2=0$$ ✓ Mathematica : cpu = 0.585785 (sec), leaf count = 395

$$\{\text{Solve}[\frac{a(-\log \left(\frac{(a^2-1)(a\sqrt{a^2-\frac{y(x{)}^2}{x^2}-1}+a^2-\frac{iy(x)}{x}-1)}{a^3(\frac{y(x)}{x}-i)}\right)+\log (-\frac{(a^2-1)(a\sqrt{a^2-\frac{y(x{)}^2}{x^2}-1}+a^2+\frac{iy(x)}{x}-1)}{a^3(\frac{y(x)}{x}+i)})+\log (\frac{y(x{)}^2}{x^2}+1))-2i{\tan }^{-1}\left(\frac{y(x)}{x\sqrt{a^2-\frac{y(x{)}^2}{x^2}-1}}\right)}{2(a^2-1)}=\frac{a\log (x-a^{2}x)}{1-a^2}+c_1,y(x)],\text{Solve}[\frac{2i{\tan }^{-1}\left(\frac{y(x)}{x\sqrt{a^2-\frac{y(x{)}^2}{x^2}-1}}\right)+a(\log (-\frac{(a^2-1)(a\sqrt{a^2-\frac{y(x{)}^2}{x^2}-1}+a^2-\frac{iy(x)}{x}-1)}{a^3(\frac{y(x)}{x}-i)})-\log \left(\frac{(a^2-1)(a\sqrt{a^2-\frac{y(x{)}^2}{x^2}-1}+a^2+\frac{iy(x)}{x}-1)}{a^3(\frac{y(x)}{x}+i)}\right)+\log (\frac{y(x{)}^2}{x^2}+1))}{2(a^2-1)}=\frac{a\log (x-a^{2}x)}{1-a^2}+c_1,y(x)]\}$$

✓ Maple : cpu = 0.736 (sec), leaf count = 229

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